Supplementary semilattice sums of rings

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Supplementary semilattice sums of rings
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Turman, Eleanor Geis, 1948-
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Rings (Algebra)   ( lcsh )
Ideals (Algebra)   ( lcsh )
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Thesis--University of Florida.
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Includes bibliographical references (leaf 52).
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by Eleanor Geis Turman.
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Typescript.
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Vita.

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SUPPLEMENTARY SEMILATTICE SUMS OF RINGS


BY


ELEANOR GEIS TURMAN






























A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1980












To my parents who gave me love and support even when

they didn't agree with me, and to my daughter Elizabeth who

withstood so admirably the trials of growing up with a

graduate student.












ACKNOWLEDGEMENTS

I would like to express my thanks and appreciation to

my advisor, Mark L. Teply, for his superior example and

generous encouragement without which I would not have become

a mathematician.

I owe a great debt to Neil White whose valuable sugges-

tions were fundamental to the development of several proofs

and whose teaching has profoundly influenced my mathematical

point of view.

I would like to thank Bruce Edwards for the many hours

of generous assistance that he gave me. I would also like

to thank the other members of my committee, Alexander Bednarek,

David A. Drake, Gerhard Ritter, David Wilson and Chuck Hooper

for their comments and suggestions.


iii













TABLE OF CONTENTS


Page


ACKNOWLEDGEMENTS....................................... iii

KEY TO SYMBOLS. ..... ........... .......... .......... v

ABSTRACT .......... ....... .............................. vi

CHAPTER 1 PRELIMINARIES............................. 1

1.1. Order Theory. ......... ...... .......... 1
1.2. Semigroups............... .............. 3
1.3. Semigroup Rings......................... 6
1.4. Twisted Semigroup Rings................ 7
1.5. Matrix Rings............................ 7
1.6. Radicals. ............................. 9

CHAPTER 2 STRONG SUPPLEMENTARY SEMILATTICE SUMS....... 12

CHAPTER 3 RADICALS OF SUPPLEMENTARY SEMILATTICE SUMS.. 23

CHAPTER 4 RADICALS OF INFINITE MATRIX RINGS........... 36

CHAPTER 5 RADICALS OF SEMIGROUP RINGS................. 42

REFERENCES..... ........................................ 52

BIOGRAPHICAL SKETCH.................................... 53












KEY TO SYMBOLS


Symbol

Z

Z
n
N

A x B

H A.
I 1

9 E A.
i E 1

f: A + B

a f(a)

1A
gof or gf

Af-



R/I

ISI


ring of integers

ring of integers modulo n

set of nonnegative integers

Cartesian product of sets A and B

product of the family of objects {Aili E I}


direct sum of the family of objects {Aili E I}


f is a function from A to B

the function f maps a to f(a)

the identity function of the set A

composite function of f and g

inverse image of the set A

is isomorphic to

factor group (ring) of R by I

cardinality of the set S












Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy

SUPPLEMENTARY SEMILATTICE SUMS OF RINGS

By

Eleanor Geis Turman

August 1980

Chairman: Mark L. Teply
Major Department: Mathematics

A generalization of the concept of a decomposition of a

ring into a direct sum of ideals is introduced. A ring R is

a strong supplementary semilattice sum of rings Ra (a Y) if

Y is a semilattice, R is the direct sum of the Ra (a E Y) con-

sidered as abelian groups, and there exists a family of

homonorphisms {a, B: Ra + R la,8 E Y, a > @} satisfying the

following conditions:

(i) aa is the identity map on Ra for each a E Y;

(ii) for a B a y ,88y L = ,y;

(iii) the product, x*y, of x e R and y E R for a,B e Y

is defined by x*y = (x aa) (yBaB) e R a where aB denotes

the product of a and B in Y. Various mapping properties of

such rings are investigated.

The question of describing the 7-radical of R for a

radical property i in terms of the r-radicals of the R

(a E Y) is investigated. If R is 7-semisimple for all a E Y,

then R is r-semisimple. The converse is false in general. We

vi







determine conditions on the semilattice Y that guarantee

that R is r-semisimple if and only if Ra is 7-semisimple for

all a E Y. If T is a strict, hereditary radical property,

then the r-radical of R is the sum of the 7-radicals of the

R (a E Y). In the case that all principal ideals of Y are

finite, then, for an arbitrary radical property 7, the

n-radical of R is the sum of ideals that are isomorphic to

the i-radicals of the Ra (a E Y). For the properties of

right quasi-regularity, G-regularity, or nil, the i-radical

can be described in terms of homomorphisms from R onto R

for a E Y.

The n-radicals of various matrix rings are determined

for a broad class of radical properties. If Tis the ring

of all bounded I x A matrices over a ring R with sandwich

multiplication and sandwich matrix P, the i-radical of T is

the set {A E TIPAP has entries in the r-radical of R}.

The above results are applied to any semigroup ring R[S3

over a completely regular semigroup to effectively reduce the

problem of determining the T-radical of RES] to the group

ring case.


vii












CHAPTER 1
PRELIMINARIES

In this chapter we introduce some fundamental definitions

and results that will be used throughout this dissertation.


1.1. Order Theory

A set P together with a relation 5 that is reflexive,

antisymmetric, and transitive is called a partially ordered

set or poset. A chain is a poset in which any two elements

are comparable. The dual poset P* is the poset obtained by

inverting the order relation on P. The order relation on

P* is denoted by <*. If P contains a unique minimal (maximal)

element, then this element is called the 0-element (1-element)

of P.

Let a,b E P. We say that b covers a or a is covered by

b if a < b and a < c < b implies a = c. If b covers a, then

we write a < b. The elements covered by 1 are called

copoints. An interval of P is any set [a,b] = {c E Paa < c b}

for a,b E P. P is locally finite if every interval in P is

finite.

If S is a subset of P, then an element a of P is a lower

bound of S if, for every b E S; a < b. Further, a is a

greatest lower bound of S if a is a lower bound of S and if

a 2 c for every lower bound c of S. If S has a greatest lower

bound, it is unique. We denote the greatest lower bound of








S by g.l.b.S. The term "meet" will frequently be used in

place of greatest lower bound. A poset P in which any two

elements have a meet will be called a lower semilattice or

simply semilattice. The concepts of upper bound, least

upper bound or "join," and upper semilattice are defined

analogously. A poset that is both an upper and a lower semi-

lattice is called a lattice. A subset S of P is called a

lower (upper) subsemilattice if S is itself a lower (upper)

semilattice and meets (joins) in S coincide with meets (joins)

in P.

A subset I of a poset P is called an ideal of P if, for

all x,y E P, the conditions x e I and y x imply y E I. For

S S P, the set {b E Plb : a for some a E S} is an ideal of P

called the ideal generated by S. Ideals that are generated

by singleton sets are called principal ideals.

Let P be a locally finite lower semilattice. The M6bius

function p:P x P Z is defined inductively by


p(a,a) = 1 for all a E P

p(a,b) = E y(a,c) = E P(c,b) if a < b
a:c
p(a,b) = 0 if a $ b


Here, c indicates that the sum is over all c that satisfy the

inequality. Consider as an example the following diagram

which illustrates the partial order on P:













0


The numbers in the diagram are the values of p(a,l) for

a E P.


1.2. Semigroups

A set S together with an associative binary multiplica-

tion is called a semigroup. A nonempty subset T of S is

called a subsemigroup if T is closed under the multiplication

in S. A subgroup of S is a subsemigroup which is also a

group under the multiplication in S. A subsemigroup T of S

is an ideal if the conditions t E T and s E S imply that

st,ts E T. Any nonempty set of ideals of S is closed under

arbitrary union and under intersection (if nonempty). The

intersection of all ideals of S containing a nonempty subset

A of S is the ideal generated by A. A principal ideal is one

generated by a singleton set. S is simple if it contains no

proper ideals (i.e., S is its only ideal).

An element z of S is a zero of S if z = sz = zs for all

s e S. A semigroup can contain at most one zero. A semigroup

with zero z is 0-simple if S and z are its only ideals. An

element e of S is an identity of S if s = se = es for all

s e S. A semigroup can contain at most one identity.
2
An element a of S is idempotent if a = a. A band is a

semigroup all of whose elements are idempotent. On any







commutative band, define the partial order a h b if and only

if a = ab. Under this ordering, S becomes a lower semilattice.

Conversely, if S is a lower semilattice then S is a commutative

band, where the product of any two elements in S is defined

to be their meet.

A mapping from a semigroup S into a semigroup T is a

homomorphism if (ap) (b) = (ab)) for all a,b e S. An equiva-

lence relation p on S is a congruence if the conditions apb

and cPd imply acpbd. If #:S T is a homomorphism, then the

relation p on S defined by "apb if and only if a4 = b4" is a

congruence on S.

A semigroup S is a semilattice Y of semigroups S (a E Y)

if there exists a homomorphism ( of S onto Y such that
-1
S = a1 for all a E Y. Note that S is then the disjoint

union of the subsemigroups S (a E Y) and that S S S p for

all a,B E Y. S is called a strong semilattice Y of semigroups

Sa (a e Y) if S is a semilattice Y of semigroups S (a E Y)
and if there exists a family of homomorphisms {1 : S SI

a,B e Y, a Z 8} satisfying the following properties:

(i) Laa is the identity map on S ;

(ii) for a t B : y, 0,) Y = ;

(iii) for a E S and b E SV, the product ab in S is

defined by

ab = (aa,) ((ba ).


Let G be a group. Let G denote the semigroup obtained

by adjoining a zero element 0 to G. Let I and A be nonempty




5


sets, and let P: A x I G be any function. A function

A: I x A G0 having exactly one nonzero value is called a
0
Rees I x A matrix over G If a E G, then (a)i will denote

the Rees I x A matrix whose nonzero value a is the image of

(i,X). The image of (X,i) under P is denoted by p i. We

define the product of two Rees I x A matrices A = (a)ix and

B = (b) j by


A o B = (ap jb)i .


Under the multiplication the set of all Rees I x A matrices

over G forms a semigroup called the Rees I x A matrix semi-

group without zero over the group G with sandwich matrix P.

This semigroup is denoted by M(G;I,A;P).

An idempotent f of a semigroup S is said to be primitive

if f 0 and if, for any idempotent e of S, the condition

e f implies e = 0 or e = f. A semigroup is completely simple

if it is simple and contains a primitive idempotent. In

particular, any finite semigroup without zero is completely

simple. A semigroup with zero is called completely 0-simple

if it is 0-simple and contains a primitive idempotent.

An important theorem due to Rees [2, Theorem 3.5] states

that a semigroup S is completely simple if and only if it is

isomorphic to a Rees I x A matrix semigroup without zero.

If sls2 = Sl, for all sls2 E S, then S is called a left
zero semigroup. If s1s2s3 = s1S3 for all sls2,s3 S, then

S is called a rectangular band. If, for any sl and s2 in S,

there exists one and only one element s3 of such that s3s1 = s2'







then S is called a left group. Left zero semigroups, rectang-

ular bands, and left groups are all examples of completely

simple semigroups.

The following assertions concerning a semigroup S are

mutually equivalent:

(i) S is a union of groups;

(ii) S is a union of completely simple semigroups;

(iii) S is a semilattice of completely simple semigroups [2,

Theorem 4.61. Such a semigroup is called completely regular.

Thus, every completely regular semigroup is isomorphic to a

semilattice Y of Rees I x A matrix semigroups M(G ;I ,A ;P )

(a E Y). If a semigroup S is a semilattice of groups, then

S is a strong semilattice of groups [2, Theorem 4.11].


1.3. Semigroup Rings

Let R be a ring, and let S be a semigroup. Then the

semigroup ring R[S] is the set of all formal finite sums of

the form E r s where r E R. Addition and multiplication
s s
seS
are defined by


r s + Z r's = Z (r + r')s
seS sES s eS

and


( E rs)( E r't) = Z r r' st = ( r r')u.
sES tES s,tES uES st=u


Under these operations RES] forms a ring.

If S is the trivial semigroup, then RES] is clearly

isomorphic to R. If a ring T is isomorphic to a semigroup








ring RE[S, where S is nontrivial, then we say that T has a

nontrivial realization as a semigroup ring.


1.4. Twisted Semigroup Rings

Let R be a ring with identity, and let S be a semigroup.

Let y be a function from S x S into the group U of central

units of R that satisfies the condition


Y(sl's2)Y(s1s2's3) = Y(s2,s3)y(slS2S3)


for all sl,2',s3 E S. We call y a twist function. The

twisted semigroup ring Rt [S] with twist function y is the

set of all formal finite sums Z r s, where r E R with
sES
addition defined by


E rs + E r's = Z (r +r')s
SES SES SES

and multiplication defined distributively by


r1s1 r2s2 = rlr2y(sls2)s1s2


for rl,r2 E R and sl,s2 E S. Note that if y(sl,s2) = 1 for

all sl',2 e S, then Rt S] is the semigroup ring RES].


1.5. Matrix Rings

Let R be a ring, and let I and A be sets. An I x A

matrix over R is a mapping A: I x A R. The value of (i,X)

is denoted by aix and is called the (i,X)-entry of A. Addition

of I x A matrices is defined in the usual manner. If A is an

I x A matrix over R and B is a A x I matrix over R, then the

product AB is defined to be the I x I matrix over R whose







(i,j)-entry is a ibj provided that this sum exists in R

(i.e., that all but a finite number of the summands are zero).

An I x A matrix A is said to be row bounded if there

exists a finite subset N of A such that a.i = 0 whenever

AX N. Column bounded matrices are defined analogously. The

set of all I x A row (column) bounded matrices over R is

denoted by M*(R;I,A) (M*(R;I,A)). The intersection
P Y
M* (R;I,A) of M*(R;I,A) and M*(R;I,A) consists of all matrices
PY p Y
which are both row and column bounded and therefore have only

a finite number of nonzero entries. Such matrices are called

bounded.

Let P be a fixed but arbitrary A x I matrix over R. If

A is a bounded I x A matrix over R, then AP and PA are column

bounded I x I and row bounded A x A matrices, respectively.

For A,B e M* (R;I,A), (AP)B and A(PB) are bounded I x A
PY
matrices, and it is easily shown that (AP)B = A(PB). We define

the product a of A and B to be


A o B = APB.


Under this multiplication, the set of all bounded I x A

matrices over R forms a ring, denoted by M* (R;I,A;P). The
py
multiplication in M* (R;I,A;P) is called sandwich multiplication,
PY
and P is called the sandwich matrix.

Let M(G;I,A;P) be the Rees I x A matrix semigroup without

zero over a group G with sandwich matrix P. Let R be a ring.

Then the semigroup ring R[M(G;I,A;P)] is isomorphic to

M* (R[G];I,A;P) in the following natrual way: Let
py







n
x = r r.s. e R[M(G;I,A;P)]. Each s. is a Rees I x A matrix
i=l
over G ; suppose s. has g. 0 as its (k.,X.)-entry and zero

elsewhere. Map r.s. onto the element of M* (R[G];I,A;P) with
S1 py
r.gi as its (ki,X.)-entry and zero elsewhere. Extend this

map linearly to all of REM(G;I,A;P)]; thus, x maps to the sum

of the images of rs.i (i = 1,2,...,n). This map is an iso-

morphism of R[M(G;I,A;P)] onto M* (R[G];I,A;P). See
PY
Weissglass [11] for details.


1.6. Radicals

Let i be a certain property that a ring may possess. We

shall say that the ring R is a r-ring if it possesses the

property fr. An ideal J of R will be called a i-ideal if J

is a T-ring. A ring which does not contain any nonzero

r-ideals will be called '-semisimple.

We shall call T a radical property if the following

conditions hold:

(i) Any homomorphic image of a ir-ring is a T-ring (i.e.,

f is homomorphic invariant property);

(ii) Every ring R contains a Tr-ideal U(R) which contains

every other r-ideal of R;

(iii) The factor ring R/U(R) is T-semisimple.

U(R) is called the f-radical of R. A r-ring is its own

u-radical, and we call it a f-radical ring.

We describe several important radicals. For details

see [3].








A. The Jacobson Radical J

An element x of a ring R is right quasi-regular (r.q.r.)

if there exists an element y of R such that x + y xy = 0.

An equivalent definition is that x E R is r.q.r. if the set

{xr ri r E R} coincides with R. A ring is said to be r.q.r.

if each of its elements is r.q.r. Right quasi-regularity is a

radical property. The Jacobson radical J(R) = {x E RIxR is

r.q.r.}. An element x of R is left quasi-regular (l.q.r.) if

there exists y E R such that y + x yx = 0. An element that

is both r.q.r. and l.q.r. is said to be quasi-regular. An

ideal of R is quasi-regular if each of its elements is quasi-

regular. J(R) is a quasi-regular ideal. For any collection

of rings {Ri.i E I}, J( I R.) = H J(R.).
iI1 il


B. The Brown-McCoy Radical G

An element x of a ring R is said to be G-regular if
n
x E G(X) = {xr r + E (r.xs. r.s.)In E N, r,ris i R}.
i=l 1 1 r 1 i i
i=l
G-regularity is a radical property. The Brown-McCoy radical

is G(R) = {x E RIG(x) = R}. For any collection of rings

{Rili E I}, G( H R.) = H G(Ri).
iEI iEI


C. The Nil Radical N

An element x of a ring R is said to be nilpotent if

there exists a positive integer n such that x = 0. A ring

R is said to be nil if every element of R is nilpotent. The

nil property is a radical property, and N(R) = {x e RjxR is

nill is the nil radical of R.







D. The Baer Lower Radical 8

A ring is said to be nilpotent if there exists a positive

integer n such that Rn={O}. The nilpotent property is not a

radical property. We define 8(R) in the following manner.

Let NO be the join of all the nilpotent ideals of R.

Let N1 be the ideal of R such that N1/N0 is the join of all

the nilpotent ideals of R/NO. In general, for every ordinal

a which is not a limit ordinal, we define N to be the ideal

of R such that N /N a1 is the union of all the nilpotent

ideals of R/Na-. If a is a limit ordinal, we define

N = u No. In this way, we obtain an ascending chain of
a ideals N N .. cN c ... If the set R has ordinal

number v, then this chain must terminate after at most v steps.

We may then consider the smallest ordinal T such that

NT = N +1 = ... This ideal N we shall call 3(R), the

Lower Baer Radical of R.


E. The Levitzki Radical L

A ring is locally nilpotent if any finite set of elements

generates a subring that is nilpotent. Local nilpotence is a

radical property. L(R) is the join of all the locally nil-

potent ideals of R and is itself a locally nilpotent ideal.

A radical property i is strict if every I-subring of a

ring R is contained in U(R); equivalently, every subring of a

I-semisimple ring is I-semisimple. A radical property n is

hereditary if every ideal of a I-ring is a w-ring. The

properties of right quasi-regularity, G-regularity, and nil

are hereditary.












CHAPTER 2
STRONG SUPPLEMENTARY SEMILATTICE SUMS

Many substantial results in the theory of rings are

concerned with a decomposition of a ring into a direct sum

of ideals. In this chapter we introduce a generalization of

direct sum decompositions.

Let R be a ring, let Y be a set, and let {Ra } Y be a

collection of subrings of R. Then E R = {r +ra +...+r I
eY 1 2 n
n E Z r e R }. R is called a semilattice sum of rings
1 1
Ra (a E Y) if Y is a semilattice, R = Z R., and Ra R8 Ra
aEY
for all a,B E Y. If R is a semilattice sum of rings Ra

(a E Y) and if, for every a E Y, Ra n E R = {0} (i.e., if

the sum is direct considered as abelian groups under addition),

then R is called a supplementary semilattice sum of rings

Ra (a E Y). We call R a strong supplementary semilattice

sum of rings Ra (a E Y) if R is a supplementary semilattice

sum of rings Ra (a E Y) and there exists a family of homo-

morphisms { p,a: Ra Rp[ a,B E Y, a > B} satisfying the

following conditions:

(i)
(ii) for a > y,
(iii) for each x e Ra and y E Rg, the product in R of x

and y, x y, equals the product (x aaB (yp, ) in R'

The concept of supplementary semilattice sum was first

introduced by Weissglass [11]. If R is a direct sum of ideals







Ia, a E Y, then R is a strong supplementary semilattice sum

of the subrings Ia (a E Y). For if a and 8 are distinct

elements of Y, then I I8 ='{0}, and therefore multiplication

on Y can be defined by aB = a0, where a0 is a fixed element

of Y. Under this multiplication, Y becomes a semilattice.

For each a E Y such that a e a0, let ,a : I + I be the
0 a,a0 a a0
zero homomorphism. For all a E Y let : I a Ia be the

identity map. Clearly, the family {0aS : I Iia 0 E Y,a > }

satisfies the conditions that make R a strong supplementary

semilattice sum of the subrings I (a E Y).

Examples of strong supplementary semilattice sums of

rings abound.


Example 2.1. Let S be a strong semilattice Y of semi-

groups Sa (a E Y), and let R be a ring. Each of the associated

semigroup homomorphisms o, : Sa S (a,B e Y, a t p) deter-

mines in a natural way a ring homomorphism from R[Sa] into

R[S ] defined by

n n
E r.s ---> r (si. 8)
i=l i=l

where r. E R and s. e S We also denote this mapping by

aB' for convenience. Clearly, this family of ring homomor-

phisms satisfies the conditions that make the semigroup ring

R[S] a strong supplementary semilattice sum of the rings

RCSa] (a E Y).


Example 2.2. Let T be any ring, and let Y be any upper

semilattice contained in the lattice of ideals of T. For







each a E Y, let R = T/a. For a,B E Y* such that a 8,

let >B: : Ra R be the canonical homomorphism. 'Then

R = E R is a strong supplementary semilattice sum of the
aEY*
rings Ra (a E Y*).

In general, a ring defined as in Example 2.2 cannot be

realized as a semigroup ring except in the trivial way. To

illustrate this assertion, we consider rings constructed by

the method of Example 2.2 in the following two examples.


Example 2.3. Let R be the strong supplementary semi-

lattice sum of the rings R Z4 and R = Z2, where the

associated semilattice is the two element chain {a > 8} and

the associated non-identity homomorphism a, : Ra -* R is

the canonical homomorphism. Then as an abelian group

R = Z4 Z2. Multiplication in R is defined by


(a,b)-(c,d) = (ac,(aa,6)d+b(c a,)+bd),


where (a,b), (c,d) E R.

R cannot be realized as a semigroup ring except in the

trivial way. For, if R were isomorphic to TES] for some

ring T and semigroup S, then as abelian groups R = Z 4 Z2

E T. Hence ISI = 1 and T = Z4 Z2 by the Fundamental
IsI
Theorem of Finitely Generated Abelian Groups.


Example 2.4. Let Y be the lattice of ideals of Z con-

sisting of Z, {0}, and all ideals generated by a prime p.

Then Y* has the following order structure:














Let R = E R be as defined in Example 2.2. Then R is a
aEY* a
strong supplementary semilattice sum of the rings Z, {0}, and

Z (p prime).

R cannot be realized as a semigroup ring except in a

trivial way. For, if R were isomorphic to T[S] for some ring

T and some semigroup S, then as abelian groups R = 0 Z R =
aeY* a
SZE T. For each prime p, let T denote the p-primary part
Isl
of T. Then under the above isomorphism the p-primary part

of TES], 0 Z T is isomorphic to the p-primary part Z of R.
Isl P
Since Z is indecomposable, then ISI = 1 and T = R.

Many supplementary semilattice sums of rings are neces-

sarily strong supplementary semilattice sums of rings. Con-

sider the following example.

Example 2.5. If R is a supplementary semilattice sum

of rings Ra (c E Y), each of which contains an identity eX

and no other nonzero idempotent, and e e e 0 for all a,8 E Y

with a 2 then we can define the following family of mappings:


Sr,: R R x > xe ,

where a,B E Y such that a 2 By the distributive law in R,

P,)8 preserves addition. For x,y E R., (xy)x,8 = (xy)eg =

x(ye ) = x(egyeg) = (xe) (yeg) = (x,8) (y ,8). Hence, $,8








preserves products. Clearly, ,'a is the identity map on
2
Ra. Now, (eae ) = (e e )(e e ) = e (e (ea e)) = ea((e e )e )

= (e ea) (ee p) = e e Since e5 is the unique idempotent

in R, eae = e Thus, if a 2t 2 y and x e R then

xap,8,y = xe e =xey = x,y For each x E R and y E R
(a,B E Y), (x4Oaa) (yO 8a) = (xea )(yea) = x(e a(ye)) =

x((ye ,)e 8) = xyea8 = xy. Thus, the family of homomorphisms

{fa,8: Ra R1 a,8 E Y, a S} satisfies the conditions that
make R a strong supplementary semilattice sum of the rings

R (a E Y).

We begin the investigation of strong supplementary

semilattice sums by considering another associated family of

homomorphisms. For the rest of this chapter we assume that

R is a strong supplementary semilattice sum of rings Ra (a E Y).

For each x E R and a E Y, we define xa to be the projection

of x onto R We define supp x = {a E YI xa e 0}. For each

a E Y, we define the mapping e: R -R R by


xa = E x p



Lemma 2.6. The family {( : R Ral a E Y} defined above

is a family of onto ring homomorphisms.


Proof. Let a E Y, and let x,y E R. Then (x+y)~a

(Ex +Ey)a = (E(x +y))c~ = E (xB+y)) = (x a+yR 5,a)


= X Or + E Ya ,a = x4a + y,' and (x'y) = ((=xH ()y) ))
P a 201 a'a P a








=( (x 8 Byv) )(yy VBy)) a = ((x EB, By) (y yyy)) $BY,








homomorphism. Since R. a = R, is onto.

It follows from Lemma 2.6 that if 7 is any homomorphic


*-ring for all a Y. In particular, if R has d.c.c. (a.c.c.)

on left ideals, then Ra has d.c.c. (a.c.c.) on left ideals
for all a Y. The converse is in general false unless Y

is finite.


Theorem 2.7. If Y is finite, then R has d.c.c. (a.c.c.)

on left ideals if and only if Ra has d.c.c. (a.c.c.) on left

ideals for all a E Y.


Proof. We first prove the result in the case when Y

has just two elements. If R = R + RV, then either aB = a = a

or aB = Ba = 8. Without loss of generality we assume aB = Ba

= B. Then R R c R and RR c R which implies that R

is ideal of R. Now R/R = R,/(R, n R ) = Ra. Hence, R has

d.c.c. (a.c.c.) on left ideals if and only if Ra and Rp have

d.c.c. (a.c.c.) on left ideals [4, Theorem 8.1.5].

Suppose IYI > 2 and assume that the result holds for

all strong supplementary semilattice sums over semilattices

with cardinality less than IYI. Let a E Y that is not the







zero of Y. Let I = RV, and let F = E R. Then I and
a t>a a
F are strong supplementary semilattice sums over semilattices

of cardinalityless than IY. Hence, I and F. have d.c.c.

(a.c.c.) if and only if R has d.c.c. (a.c.c.) for all p e Y.

Clearly, I is an ideal of R and R/I = F Hence, R has

d.c.c. (a.c.c.) on left ideals if and only if I and F have

d.c.c. (a.c.c.) on left ideals. The result follows by

induction.

We next consider a fundamental mapping property of strong

supplementary semilattice sums.


Theorem 2.8. Let R and T be strong supplementary semi-

lattice sums of rings {R }aEY and {T )IY, respectively, over

the same semilattice Y. Let {() a: R -- RIa,B e Y, a B}

and {( :R R la E Y} be the families of homomorphisms associ-

ated with R. Let {, : Ta ToIa,B e Y, a B} and

{a : T T aa E Y} be the families of homomorphisms associated

with T. Let {f : Ra T la E Y} be a family of homomorphisms

making the following diagram commute for all a E Y.




f fa
v i v
T > T
a e


Then there exists a unique homomorphism 8: R T making the

following diagram commute for all a E Y.








^ a
R --> R


e f

T ----> T


Proof. By definition of the direct sum of abelian

groups [4, Theorem 1.8.5], the mapping 8 defined by

x9 = x a a is the unique additive homomorphism making the
aEY
following diagram commute for all a E Y:


R > R = Z R
a I eY a

f e
v v
T > T = ZT
a Y.. a
aeY

Let x,y e R. Then (x y) = (( Z xa)-( Z y))
aEY aEY
c (x~laB, a)(ys#B,aB))E = ((xa=a,aB) (yB aB))faB =

a, EY Ct, EY

a (xt oaac a1 f Ot f (Xaxf a pa') (Ytfa1Pf0 ar ) a=
a,BY a, EY

S(x fa )(yf) = ( E x f ).( Z y f ) = (xe)*(ye). Hence
a,B'Y aEY aEY

e preserves multiplication.

Let x E R. Then x9-a = ( Zx xf)a = Zx f ga
BEY (ka

EB f = (Z Ex )fa = (x a)f Hence, a = f f

for all a E Y.

Suppose that g: R T is a homomorphism such that

g9a = 4faa for all a E Y. Let a E Y and let xa e Ra. If

xag = 0, then 0 = 0a = x ~a = xa f = x f and hence






x g = x f If x g O, then let B be maximal in supp x g.

If B a, then (xCg)8 = (xg)8 6 = xa fg = Of = 0, which is

a contradiction. Hence, 6 a. If B < a, then 0 = (x g)a

x ga = a fa = xafa; thus (x) B = x UgB = x Bf =

xaBf = x fa a, = Oi,B = 0, which is a contradiction.
Hence B = a, and (xg)o = x, Ug = x fa = x fa. Suppose

there exists y E Y such that a covers y in supp x g. Then

xa a,yfy = xa yfy = xa ay = (xg)y + (x g) ay = (x g) +

(xa f ) y = (xag) + xa arfy; thus (x g) = 0, which is a
contradiction. Hence supp x g = {a}, and xa f = (x g)a = x g.

Thus, for all a E Y, the following diagram commutes:

R -> R


a I
V V
T > T


But a is the unique homomorphism with this property. Hence

e = g.

Theorem 2.9. If {f : Ra T |a E Y} is a family of

one-to-one homomorphisms (onto homomorphisms), then e is

one-to-one (onto).


Proof. Suppose that fa is one-to-one for all a E Y.

Let x,y E R such that x6 = ye. Then, for all a E Y, x0 f =

xe a = yBeO = yo f Since fa is one-to-one, xo = y a for

all a E Y. Let P = supp x u supp y, and let a be maximal in

P. Then x = x00 = yoc = ya. Let e P such that B is not

maximal in P. Assume for all y > B, Y P, that x = y.







Then xg = x + E x Z x = x y 4Y =
S> Y> Y Y>,

Yg8 Y yyy = YB. By induction, x8 = y8 for all 8 E P,

and hence x = y. Thus e is one-to-one.

Suppose that fa is onto for all a E Y. Let y E T. Then,

for each a E supp y, there exists xa E Ra such that xaf = ya.

Let x = x. Then xe = E x f = E = y.
aesupp y aEsupp y aesupp y
Hence, & is onto.

It follows from Theorem 2.9 that R and T are isomorphic

if and only if Ra and T are isomorphic for all a E Y.

The following result will be used in applications to

semigroup rings discussed in Chapter 5.


Lemma 2.10. Let R be a strong supplementary semilattice

sum of rings Ra (a E Y) with associated family of homomorphisms

{O, a: Ra + RN'a,B e Y, a 2 S}. For each a E Y, let Ta be a

ring isomorphic to R Then T = T is a strong supplemen-
aeY
tary semilattice sum that is isomorphic to R.


Proof. For each a E Y, let f : Ra Ta be an isomorphism

onto T Let a,B E Y such that a > and let x e T Define
-1
the mapping a ,B: Ta Tg by xipB = yBfs, where y = xf1 .

Since fa is an isomorphism, a,B is well-defined. For each

x,x' e Ta, x1g, + x'v ,a = Y4,Uefg + y'cPaf5 =

(y + y') a,c f = (x + x')a,8, and (xa,a) (x'ia,B)

(ya,B f ) (y',f) = (yy')a,f = (xx')pa,B. Hence, Wa,B

is a homomorphism. For x E Ta, xaa = Y,af = = y x,

which implies that aa is the identity map on Ta. Let
aXflXa-







a,B,y E Y such that a 8 2 y, and let x E Ta. Then

x,6(,Y = (yt,Bf) ', = Ya,By = yaryfy = xaY-
Hence, T = Z T is a strong supplementary semilattice sum.
aeY
By Theorem 2.8, there exists a homomorphism 8: R T.

By Theorem 2.9, 6 is an isomorphism onto T.

The following result will prove useful in the investiga-

tion of the T-radical of R.


Theorem 2.11. R is isomorphic to a subdirect product of

the collection {Ra } EY


Proof. Let Ka denote the kernel of cc (a E Y). Suppose

there exists 0 r x E n K Then supp x < 0. Let 8 be maxi-
aEY
mal in supp x. Since x e KV, then xa = x4 = 0, which is a

contradiction. Hence, n K = {0}, and R is isomorphic to
YEY
a subdirect product of rings Ra (a e Y) by [4, Exercise 9.3.1].











CHAPTER 3
RADICALS OF SUPPLEMENTARY SEMILATTICE SUMS

Let ir be a radical property. Let R be a supplementary

semilattice sum of rings Ra (a E Y). We seek to describe

the T-radical of R in terms of the T-radicals of the R

(a E Y).

Teply et al. [8] has shown that, for a hereditary pro-

perty n, R is r-semisimple whenever Ra is i-semisimple for all

a E Y. The converse is in general false; in fact, for any

nontrivial semilattice Y, there exists a Jacobson semisimple

ring which is a supplementary semilattice sum of rings not all

of which are Jacobson semisimple [9].

We show that if R is a strong supplementary semilattice

sum of rings R (a E Y) and if n is any radical property (not

necessarily hereditary), then R is 7-semisimple whenever R

is 7-semisimple for all a E Y. If R is r-semisimple, then not

all the R need be r-semisimple [8]. We determine conditions

on the semilattice Y which guarantee that R is fT-semisimple

for all a E Y if R is T-semisimple. This gives an answer to

Question 1 of Weissglass [11] for the strong case.

In certain cases, the r-radical of R is the sum of the

n-radicals of the R (a E Y). Gardner [5] has shown that this

is the case whenever T is strict and hereditary and R is a

supplementary semilattice sum of rings Ra (a E Y) for a finite

semilattice Y. We show that the condition that Y be finite

can be dropped.







In the case that the sum is strong and all principal

ideals of Y are finite, then, for an arbitrary radical pro-

perty n, the r-radical of R is the sum of ideals that are

isomorphic to the r-radicals of the R (a E Y).

If n is the property of being right quasi-regular,

G-regular, or nil, and R is the strong supplementary semi-

lattice sum of the R (a E Y), then the r-radical of R can

be described as the set {x E R x#a E U(R ) for all a E Y}.

In this case, R is a i-ring, if and only if R is a i-ring

for all a E Y.

Throughout this chapter we let R be a strong supplementary

semilattice sum of the rings Ra (a E Y), unless otherwise

specified. Let { ,8: R c R lJa, E Y, a a} and

{( : R R la E Y} be the associated families of homomorphisms.

Since R. = R0a is a homomorphic image of R, it follows that,

for all a E Y and x e U(R), x#a e U(R ). We thus obtain the

following result.


Theorem 3.1. If R is i-semisimple for all a E Y, then

R is i-semisimple.


Proof. Let 0 t x e U(R), and let a be maximal in suppx.

Then x = xca E U(R ). Since U(R) = 0, then x 0, which

is a contradiction.

If R is i-semisimple, then not all of the R need be

i-semisimple; a problem may occur if R has a nonzero i-ideal

and there are infinitely many 8 E Y closely beneath a in the

order > of Y. We consider two conditions on Y and show that







they guarantee that R is 7-semisimple if and only if R is

iT-semisimple for all a E Y.

A semilattice Y is called an m.u. semilattice if

(i) for every a E Y, the set {B E YjI < a} is finite;

and

(ii) if a,y e Y with y < a, then there exists 8 e Y such

that y -< < a.


Lemma 3.2. Let Y be a semilattice with zero z. If A is

an ideal of R then A is an ideal of R.


Proof. Let x e A, let a E Y, and let y E Ra. Then

x y = x(y ,z) E A and y x = (yaz )x A.

The following results will be used in the proof of

Lemma 3.5.


Proposition 3.3. Let P be a finite semilattice. If

a,B E P with a < 8, then E ((a,y) = EZ (y,B) = 0.
a:
Proof. Z P(a,y) = Z y(a,y) + P(a,) = -p(a,S) +
a
(a, ) = 0.


Proposition 3.4. Let P be a finite semilattice with

copoint set C. If rk denotes the number of k-sets A c C
k
with g.l.b. A = 0, then p(0,1) = E (-1) rk [1, Prop. 4.4.3].
k20

Lemma 3.5. Let Y be an m.u. semilattice, and let a E Y

such that a is not a zero of Y. Let P be the subsemilattice

generated by the set {a) u {B e YIB < a}. Let p be the







Mobuis function on P x P. Then the mapping *: R -+ R defined

by


x* = Z (Ba)x
BeP


is a one-to-one homomorphism that maps ideals of R onto

ideals of R.


Proof. Let x,y E R Then (x + y)* = E p(B,a)(x+y)4, -
aeP
SFi(B,a) (xA 8 + Y'c 8) = p(,aC)x + E P(,ca)y p =
BEP 86P EP
x* + y*. Hence preserves addition.

Let I be an ideal of R The proceeding paragraph shows

that I* is closed under addition. Let x E I, let y e Y, and

let y E R We will show that y x* e I*.

If y > a, then y x* = y Z p(8,a)x =a
BEP
P B(B,a)y x-,XB = 1(,a)(y y, ) (x ,B ) =
BEP BEP
E P(U,a) (y a, ) (x) ) = E (6,a) ((yP ?)x) p =
EP EP'
((y, )x)* e I*.

If y < a, let C = {6 e YI6 < al. For a nonempty subset

c of C, let c denote the product in Y of the elements of c,

For the empty subset c of C, let c = a. Let A = {( e Cy 5 6}

and let B = {6 E Cjy 6}. For each 8 e P, let r ,k denote

the number of k-subsets c of C such that c = 0. By Proposi-

tion 3.4, p(B,a) = r (-1) rk for each B E P. Hence
kO
x* = E p(,a)x = E (-1) Bk xr ,c If c is a
$eP ai, -EP k20 rk ,
subset of C, then c = a u b for some a cA and b c B. Hence






laubi


x* = E E (-1)krg kx,, = E (-1)
BEP k20 acA
bcB


y.x* = y E (-1)
acA


E (-1)
acA
bcB


laubi


xa -a = E (-1)
',ab acA
o _


bcB
laubi
(Y,yab ca ab


xaaEbU


Thus


laubi
y-x a, a


For each a c A, a 2 y; hence yab = yb. Thus

y.x* = 2 (-1) aub (y 'ya) (x ,5) =
acA ) (y,yab cX,yab
acA
bcB


= E (-1)
acA
bcB


laubl


(ycp-) (xo -).
YIYb cx,yb


Since Y is an m.u. semilattice, A # 0. Hence E (-1)
aub acA


(y) (xXa, =
aubjb acAb


It follows that yxx* = Z (-1)
acA
bcB


E [ E (-1
bcB acA


E [o0(-1)
bcB


= 0.


Ib|
](-1) (y ,y) (x y) =


)


Sbl
(yd ) (xc yE, ) = 0 e I*.


If a and y are unrelated (i.e., if neither y > a nor

y < a holds), then y-x* = y pi(B,a)x,8 =
BEP

P p(8,ac)y xa, = E P(B,a) (y y ) (xxB) =
(y ) ,)x = (y ) x*EPince


Z PiS',a) (yo yya OyOC'X Y) (xCPc ~Ya = :va)(i( ,cx)y(xO ,*(XO
SEP 'EP


(yO yy) 21i(S,)xopcx = (Yxyyy) x*. Since ya < a,
B EP'


bcB







it follows from the preceding paragraph that y x* =

(yy,y) x* = 0 E I*.

We have shown that I* is closed under left multiplication

by elements of R for each y e Y. Since I* is closed under

addition, it follows from the distributive property of R

that I* is closed under left multiplication by elements of R.

A symmetric argument shows that I* is closed under right

multiplication by elements of R. Thus I* is an ideal of R.

Let x,y E R,. Then x* y* = ( Z (B,,a)x4 B) y* =


Z ((,')x Y* = (aa)x*, y* + Z i( ,y)xeB Y* =
B i e B

x y* + Z y(8,a)x ,B y*. As shown above, x y* = (xy)*


and x~aB y* = 0 for all 8 < a. Hence, x* y* = (xy)*,

which implies that preserves multiplication.

Let x and y be distinct elements of R Then (x*)f =

x y = (y*) a. Hence is one-to-one.


Theorem 3.6. Let Y be an m.u. semilattice. Then R is

i-semisimple if and only if R is I-semisimple for all a E Y.


Proof. For all a E Y, x e U(R) implies that xOa E U(R )

since A is an onto homomorphism. Thus, if R is r-semisimple

for all a E Y, x4{ = 0 for all a e Y; hence x = 0, and R is

T-semisimple.

If R contains a nonzero r-ideal I for some a E Y, then

by Lemma 3.2 or Lemma 3.5 I* is a nonzero n-ideal of R. Thus

if R is T-semisimple, then each Ra is also r-semisimple.







Teply et al. [81 has shown that the condition that Y be

an m.u. semilattice cannot be dropped from the hypotheses of

Theorem 3.6.

We next determine the r-radical for a particular type

of m.u. semilattice. Let Y be a semilattice in which all

principal ideals are finite. Then Y is locally finite. Let

p be the M6bius function on Y x Y. Let a e Y, and let P be

the subsemilattice of Y generated by the set {a} u {B E Yj

B < ~ al. Let p be the M6bius function defined on P x P

Let y E P such that y < a. If C denotes the set of copoints

of a above y, let rk denote the number of k-sets A c Ca with

g.l.b.A = y. Then p(y,a) = E (-1) rk = Pa(y,a) by Proposi-
k20
tion 3.4. Let y e Y P with y < a. Then y(y,a) = 0 El,

Theorem 4.301. It follows that for any function f: Y R,

E p(y,a) (y)f = p a (y,a) (y)f.
y:a yEa
yEY yEPa

In the following Theorem, for a given a E Y, the homomor-

phism from R into R defined in Lemma 3.5 will be denoted by

*a. Thus, for x e R x = Z p(B,a)xb ,.
aEP


Theorem 3.7. Let Y be a semilattice in which all principal
*a
ideals are finite. Then R = E Ra 9 E R
aeY a aEY a


Proof. Note that Y is necessarily an m.u. semilattice.

Hence for all a E Y, R is an ideal of R by Lemma 3.2 and
a *B *a
3.5. We note that R n R = {0}. For, if x E R
a 13a a
Bti







then a is the unique element of Y that is maximal in supp x,
*B
but a cannot be maximal in supp y for any y E R Hence
a
Z Ra is a direct sum.
aEY a

Let x e R. Let j and 6 be the M6bius and Delta functions

on Y x Y respectively. Then x Z = Z 6(y,8)x ,Y
EY Y<8

= Z [ Z v(Y,a)1]XBpY = (y,a)xp
Yas yaseB yRae R

= Z Z x Z u(y,a)xfo(,y] = Z [ Z (ya)xr al a( y
a 6a y
*. *


a 62a a Bxa a

*a *a
R = Z R By Lemma 3.2 and Lemma 3.5, R a R for all
aE a a a
aEY

a E Y; hence R = Z R Z R
aEY aEY a


Corollary 3.8. Let Y be a semilattice in which all

principal ideals are finite. Then U(R) = a Z U(R ) a
aEY

{x e Rnlxa E U(R ) for all a E Y}.


Proof. R = Z R a by Theorem 3.7. Hence, e Z U(R )*a
aEY a aY a

is an ideal of R. Since U(R ) is isomorphic to the xf-ring

U(R ) for each a E Y by Lemma 3.5, then $ Z U(R ) a c U(R).
aEY
Let x E U(R). Then, since R = R(( is a homomorphic image of
aa a
R, then xa E U(R ) for all a E Y. Thus, (x4 ) E U(R )*a

for all a E Y. It follows from the proof of Theorem 3.7 that

x = Z (x ) E Z U(R ) Hence, U(R) c 0 Z U(R )X .
aEY a Y aeY

Thus, U(R) = Z U(R ) = {x E Rjx a E U(R ) for all a E Y}.
aEY a







Corollary 3.9. Let Y be a semilattice in which all

principal ideals are finite. Then R is a i-ring if and only

if Ra is a T-ring for all a E Y.

Now, let T be a strict, hereditary radical property,

and let R be a supplementary semilattice sum of rings Ra

(a E Y) which is not necessarily strong. Gardner [5] has

shown that Z U(R ) is a T-ideal of R and that R/( E U(R ))
aeY aeY
Z R /U(R ), where the latter sum is a supplementary semilattice
aEY a
sum. Hence, E U(RQ) c U(R), and R/ E U(RQ) is r-semisimple
aeY aEY
since it is isomorphic to a supplementary semilattice sum of

r-semisimple rings (Ra/U(Ra) (a E Y) and is therefore r-semi-

simple [8]. We have proved the following result:


Theorem 3.10. Let i be a strict, hereditary radical

property and let R be a supplementary semilattice sum of rings

R (a E Y). Then U(R) = U(R ).
aEY
This answers in the affirmative the question of whether

or not hereditary strict radicals commute with formation of

supplementary semilattice sums posed by Gardner [51.

For a general radical property r, E U(R ) need not even
aEY
be an ideal of R. We will now determine the I-radical of R

for various radical properties i in the case where R is a

strong supplementary semilattice sum of rings Ra (a E Y).

We have seen that the condition on an element x of R

that x~a E U(R ) for all a E Y is necessary for x E U(R). For

an arbitrary radical property T, it is not sufficient. However,







if w is the property of right-quasi-regularity, G-regularity,

or nil, we shall show that this condition is both necessary

and sufficient.

Let x denote the image in the direct product T = I R
CEY
of x E R under the embedding of R into T as a subdirect pro-

duct R (see Theorem 2.11).


Theorem 3.11. J(R) = {x e Rjx4 E J(R ) for all a E Y}.


Proof. Let x E R such that xcp E J(R ) for all a E Y.

We will show that x E J(R). Since x H J(R ) = J( R ) =
aeY a aEY
J(T) [4, Theorem 2.17], then T = {xy yj y E T} [3, p. 913.

Let z e R. Then there exists y E T such that z = xy y.

Hence, for all a E Y, zpc = (xO )ya Y. We will show that

y E R.

Let P be the subsemilattice of Y generated by suppx u suppz.

Then P is finite. Let p be the M6bius function on P x P. For

each a E P, let r = E w(a,B)yp Q ,, where the sum is over
a 3 p pF3a3
S e P. Then r = r E R.
aeP


For each a E Y, r~a = r rQa Z [ Z (,Y)y BY"


E [ E yj(By)yQ A ~B ]= Z' Z v(fY)y Y = Za P(S,Y)y CP
e>a yS@ S'ca y> ay < 1y

cc [ Z (3Y) Yy,ca = P(a',a)ccy A = Y" Hence, r and y
a
agree componentwise, and thus y = r e R.

Since z was an arbitrarily chosen element of R, it follows

that R = {xy yj yE R}. Thus x e J(R), and hence x e J(R).


,a







Corollary 3.12. R is a quasi-regular ring if and only if

R is a quasi-regular ring for all a E Y.


Theorem 3.13. G(R) = {x E Rx a E G(R ) for all a E Y}.


Proof. Let x E R such that xa E G(R ) for all a E Y.

We will show that x e G(R). Since x E I G(R) = G( R) =
aeY aeY
n
G(T) (see Section 5 of Chapter 1), then T = {xy y + E (y.xz -
i+l
YiZi) In E N; y,yi,zi E T} by [3, p. 116]. Let r E R. Then

there exists n E N and y,yi,zi E T for i = 1,2,...,n such that
n
r = xy y + (yxzi yizi). As in the proof of Theorem
i=l
3.11, there exist w,wi,t. E R for i = 1,2,...,n such that

w= y, wi = i and ti = zi for i = 1,2,...,n. Hence, Y,yi,

z. E R, which implies that x e G(R). Thus x e G(R).


Corollary 3.14. R is G-regular if and only if Ra is

G-regular for all a E Y.


Theorem 3.15. N(R) = {x E R jx4 E N(R ) for all a E Y}.


Proof. Let x E R such that x a E N(R ) for all a E Y.

Let z E R, and let P be the subsemilattice of Y generated by

supp x u supp z. Then P is finite. Since xea E N(R ) for

each a E P, then each x a (a E P) generates a nil right ideal

of Ra [3, p. 18]. Hence there exist integers na for each
na
a E P such that ((x )(z a)) = 0. Let n = max n For
aEP
SE Y, let E denote the meet of the set {B E Pl 2 al. Then














n
[(xc) (z( )])n = [(Z n xB(@p ) ( 2 z@ B, )] =











Since a was an arbitrarily chosen element of Y, the ath com-

ponent (xg UH(zp) of xz raised to the power n is zero for all
n n
a E Y. Hence, (xy) = 0. Since z was an arbitrarily chosen

element of R, x generates a nil right ideal of R. Hence,

x e N(R), which implies that x E N(R).


Corollary 3.16. R is a nil ring if and only if R is a

nil ring for all a e Y.

In certain cases, for the properties right quasi-regular,

G-regular and nil, U(R) = E U(R). We first prove the
OcEY
following preliminary result.


Lemma 3.17. Let R be a strong supplementary semilattice

sum of rings R (a E Y) with associated family of homomorphisms

{Q,6: Ra R-Ia,6 e Y, a > 3}. If a,B is onto for all

a,B e Y with a 2 3, then U(R) E Z U(R ).
CEY

Proof. Let x E U(R). Since R = Rp$ for all 6 e Y, then

xp e U(R ). Let a be maximal in supp x. Then x = xp e U(R ).

Let E supp x such that 6 is not maximal in supp x. Assume

that, for all y e supp x such that y > p, we have x E U(R ).







Then x+ = xx + x 7 x # = Since
>B Y 7,8 Y> Y> 7>3 Y ,

A, is onto, x yya E U(R ) for all y > 8. Hence,

x = x Y Ex yy, E U(R ). The result follows by induction.



Theorem 3.18. If U = J, G, or N, and if (,8 is onto for
--a,
all a,8 E Y with a 8, then U(R) = Z U(R ).
aEY


Proof. U(R) _c U(R ) by Lemma 3.17. Let x E U(R )
aEY
and let y E Tg for some a,B e Y. Since (,aB is onto,

x#,aaB e U(R a) and hence x o y = (x aae) (ya ) E U(R )
and y o x = (yp 8) (xja ) e U(R ). Thus E U(R ) is an
cEY
ideal of R. Since E U(R ) is also a strong supplementary
aeY
semilattice sum of w-rings, then by Corollaries 3.12, 3.14

and 3.16, E U(R ) is a i-ideal. Hence, E U(R ) c U(R).
aeY aeY













CHAPTER 4
RADICALS OF INFINITE MATRIX RINGS

Let R be any ring. It is well known that for many

radical properties U(Mn(R)) = Mn(U(R)). In this chapter we

obtain a similar result for infinite column (row) bounded

matrices over R. We then determine the n-radical of the ring

of all bounded I x A matrices over R with sandwich multipli-

cation, where I and A are possibly infinite sets and T sat-

isfies certain conditions.

Throughout this chapter we assume the following conditions

on the property T.

(i) i-semisimple rings contain no nonzero one-sided

In-ideals.

(ii) Left ideals of w-rings are i-rings.

(iii) The i-radical of R contains all ideals A such that

AR = 0 or RA = 0.

(iv) If A is an ideal of a ring R such that R/A and A

are i-rings, then R is a i-ring; that is, i is closed under

extensions.

The five radical properties listed in chapter 1 satisfy

these conditions.

For any property T satisfying the above conditions, the

following results hold.







Result 4.1. [10] If R is a left ideal of a ring T and

if A is a i-ideal of R, then RA c U(T).


Result 4.2. [10] If R is a left ideal of a ring T and

A is a i-ideal of T, then A n R c U(R).


Result 4.3. (a) Condition (i) is equivalent to the

condition that U(R) contains all one-sided r-ideals.

(b) In the presence of conditions (ii) and (iv), con-

dition (iii) is equivalent to the condition that U(R) contains

all nilpotent ideals.


Proof. (a) Let I be a one-sided i-ideal of R. Then

the image of I under the canonical homomorphism from R onto

R/U(R) is a one-sided I-ideal of R/U(R). Since R/U(R) is

ir-semisimple, then I c U(R) by condition (i). Hence, con-

dition (i) implies that U(R) contains all one-sided ir-ideals.

The converse is clearly true.

(b) Let T satisfy conditions (ii), (iii) and (iv). Let
2
A be an ideal of R such that A = 0. Then A c U(A) by con-

dition (iii). Hence, A is a r-ring by condition (ii). It

follows that A E U(R). Assuming that U(R) contains all nil-

potent ideals of index less than n (n > 2), let A be a nil-

potent ideal of R of index n. Then, by assumption, A2 U(R).

Since (A/A2 )2 = 0, A/A2 is a i-ring by conditions (iii) and

(ii). By condition (iv), A is a ir-ring. Hence, A E U(R).

By induction on n, U(R) contains all nilpotent ideals.

Let i satisfy conditions (ii) and (iv), and suppose that

U(R) contains all nilpotent ideals. Let A be an ideal of R







such that AR = 0 (or RA = 0). Then A2 c AR = 0 (or

A2 c RA = 0). Hence, Ac U(R).

Patterson [6] has shown that J(M*(R;I,I)) = M*(J(R);I,I)).
Y Y
We generalize this theorem for radical properties satisfying

conditions (i) (iv) in the following result.


Theorem 4.4. U(M*(R;I,I)) = M*(U(R);I,I).
Y Y

Proof. For each i E I, let R. = {A M*(U(R);I,I) Ia =0
1 y rs
if r i}. Then R. is a right ideal of M*(R;I,I). Let
1 y
N. = {A E R.ia.. = 0}. Then N. is an ideal of R. such that
1 111 1 1
N.R. = {01. By condition (ii), N. is a i-ring. Since

R./N. = U(R), then R. is a I-ring by condition (iii). Hence

R. is a right T-ideal of M*(R;I,I). By Result 4.3,

R. c U(M*(R;I,I)). Since U(M*(R;I,I)) is a right ideal of
1 Y Y
M*(R;I,I) which contains the right ideals R. (i E I), then
y 1

Z R. c U(M*(R;I,I)). Hence, M*(U(R);I,I) = E R.c U(M*(R;I,I)).
iI Y Y iI Y
iel iel 1
The reverse containment was proved by Weissglass in [11].

We now determine the radical of M* (R;I,A;P).
PY

Theorem 4.5. U(M* (R;I,A;P)) = {A E M* (R;I,A;P)|
Qy pY
PAP e M*(U(R);A,I)}.


Proof. Let T = M*(R;I,I), and let S = M* (R;I,A;P).
Y PY
Define the mapping c:S T by Aq = AP. For A,B e S,

(A + B)( = (A + B)P = AP + BP = A4 + Bp, and (A o B)) =

(APB)P = (AP)(BP) = (A()(Bf). Thus $ is a ring homomorphism.

For A e S and B E T, B(Af) = B(AP) = (BA)P = (BA) e SP.

Hence Sq is a left ideal of T.







Let K denote the kernel of b. For A E K and B E S,

A o B = APB = OB = 0. Hence K o S = 0. By condition (ii),

K c U(S). Since c is a homomorphism, U(S)4 S U(S(). Con-

versely, let H be the ideal of S such that U(Sp) = H/K. Then

U(Sp)/U(S)c = (H/K)/(U(S)/K) = H/U(S) S S/U(S), which is

i-semisimple. Hence U(S()/U(S) is both w-radical and

w-semisimple. Therefore U(Sf) = U(S)p.

Let A E U(S). Then A( E U(S)# = U(Sf). For B E S,

BPAP = (Bf) (A4) E (S() (U(Sf)) SE U(T) by Result 4.1. By

Theorem 4.4, BPAP e M*(U(R);I,I).
Y
For r E R, let rE. denote the I x A matrix whose (i,X)

entry is r and whose remaining entries are zero. Then

(rEi )PAP e M*(U(R);I,I). The nonzero entries of this matrix
Y
occur in the ith row. If b j is the (X,j) entry of PAP, then

rb j is the (i,j) entry of (rE i)PAP. Thus Rbj 5 U(R) for

all X,j. Let J, j be the left ideal of R generated by bj.

Then RJ.j + RJ jR 5 U(R). Hence the image of JXj + Jj R

under the canonical homomorphism from R onto R/U(R) lies in

U(R/U(R)) by condition (ii). Since R/U(R) is 7-semisimple,

Jj + Jj R c U(R). Thus b j e U(R) for all X,j. Therefore,

PAP E M*(U(R);A,I).
Y
Conversely, suppose that PAP e M*(U(R);A,I). Then, for

all B E S, BPAP E M*(U(R);I,I) = U(M*(R;I,I)) by Theorem 4.4.
Y Y
Thus (B o A)( = (BPA)P e S4 n U(M*(R;I,I)) E U(S>) by
Y
Result 4.2. Since U(Sf) = U(S)# and U(S) 2 K, then

B o A e U(S). Let J be the left ideal generated by A. Since

S o J + S o J S c U(S), the image of J + J o S under the







canonical homomorphism from S onto S/U(S) is contained in

U(S/U(S)) by condition (ii). Since S/U(S) is w-semisimple,

then J + J o S c U(S), and thus A E U(S).


Corollary 4.6. M* (R;I,A;P) is 7-semisimple if and only
PY
if R is T-semisimple and PAP = 0 implies A = 0 for all A E M* .
PY

Proof. If A E U(M* (R;I,A;P)), then PAP E M*(U(R);A,I)
PY Y
by Theorem 4.5. If R is r-semisimple, then PAP = 0. Hence

A = 0 by hypothesis.

Conversely, suppose that S = M* (R;I,A;P) is 7-semisimple.
PY
If PAP = 0 for some A E S, then, for all B,C e S, B o A C =

BPAPC = BOC = 0. Let J be the ideal of S generated by A.

Then S o (J o S) = 0, which implies that J o S c U(S) = 0

by condition (ii). Again by condition (ii), J s U(S) = 0.

Thus A = 0.

Let r E U(R). Then P(rEi )P e M*(U(R);A,I). Hence, for
iA y
all B,C E S, B o rEi, o C = BP(rE.i)PC E M* (U(R);I,A;P) =
ixA ix PY
U(S) = 0. By the argument in the previous paragraph,

rE = 0, and hence r = 0. Thus R is r-semisimple.

The following definition is due to Weissglass [11]:

Let P be a A x I matrix over a ring T, and let

R = M* (T;I,A;P). P is cancellable with respect to R if, for

A R, A ; 0 implies AP ; 0 and PA 0.

Weissglass11, Theorem 3.7] has proved the following

result: Let R = M* (T;I,A;P). Then R is T-semisimple if and
DY
only if T is r-semisimple and P is cancellable with respect

to R.




41


Thus, in the case where T is i-semisimple, the condition

on A E R that A 0 implies PAP t 0 is equivalent to the

condition that P is cancellable.

Corollary 4.6 generalizes a result due to Munn [2,

Theorem 5.193 who considersthe case where I and A are finite

and T is an algebra over a field.












CHAPTER 5
RADICALS OF SEMIGROUP RINGS

In this chapter we apply some of the results of the

preceding chapters to semigroup rings. We assume unless

otherwise specified that n is a radical property satisfying

conditions (i) through (iv) of Chapter 4.

The class of completely regular semigroups coincides

with the class of semigroups which are unions of groups.

The main purpose of this chapter is to study the r-radical

of a semigroup ring over a completely regular semigroup. If

S is a completely regular semigroup and R is a ring, then

R[S] is a supplementary semilattice sum of semigroup rings

over completely simple semigroups. Any semigroup ring over

a completely simple semigroup is isomorphic to a matrix

ring of all bounded I x A matrices with sandwich multiplica-

tion over a group ring. Hence, the results of the preceding

chapters may be applied to reduce the problem to the group

ring case. We obtain analogous results for supplementary

semilattice sums of semigroup rings over completely 0-simple

semigroups.

Let S be a completely simple semigroup. By the Rees

Theorem [2, Theorem 3.5], S is isomorphic to (and hence we may

assume it to be) a Rees I x A matrix semigroup M(G;I,A;P)

over a group G, with A x I sandwich matrix P. Let R be a ring.







Then the semigroup ring R[S] is isomorphic to (and hence we

may assume it to be) the ring of all bounded I x A matrices

M* (R[GI;I,A;P) over the group ring REG], with sandwich
YP
matrix P [101.


Theorem 5.1. Let S = M(G;I,A;P). Then U(R[S]) = {A E R[SII

PAP e M*(U(R[G]);A,I)}. Furthermore, R[S] is r-semisimple if

and only if REG] is 7-semisimple and, for A E R[S], PAP = 0

implies A = 0.


Proof. The result follows from Theorem 4.5 and

Corollary 4.6.

In the case that the entries of P = (pli) are the

identity e of G, then for A = (a i) e R[S], the (X,i)-entry

of PAP, is Z E p.a. p = Z ea. e = Z E a We
PeA jel A3 3 P EA jIl ynEA jlI

thus obtain the following result.


Corollary 5.2. Let S = M(G;I,A;P), where P = (pXi) and

pi is the identity of G for all X E A and i E I. Then
U(R[S]) = {A = (a E) R[S]I E ai E U(R[G])}.
iA iEI
XEA

In particular, if S is a rectangular band, then G is

trivial; it follows from Corollary 5.2 that U(R[S]) =
n n
{ i r.s. e R[S]| Z ri e U(R)}. This generalizes several
i=li=l
results of Quesada [7] who determined the Lower Baer radical

of RES] in the following cases: (1) S is a left zero semi-

group and R is commutative with identity, and (2) S is a left

group and R is a ring with identity.







Theorem 5.1 permits the application of any of the results

for group rings to obtain corresponding results for semigroup

rings. For example, we obtain the following corollary.


Corollary 5.3. Let S = M(G;I,A;P), where G is a finite

group. Let R be a field. Then R[S] is Jacobson semisimple

if and only if the characteristic of R does not divide the

order of G and PAP = 0 implies A = 0, for A E RES].


Proof. By Maschke's Theorem [4], REG] is Jacobson

semisimple if and only if the characteristic of R does not

divide the order of G. The result follows from Theorem 5.1.

Now, let S be a completely regular semigroup. Then S is

a semilattice Y of completely simple semigroups S (a E Y).

For each a E Y, S = M(G ;I,A ;Pa ) without loss of generality.

Let R be a ring. Then RES ] = M* (REG ];I ,A ;P ) for each

a E Y without loss of generality, and R[S] is the supplementary

semilattice sum of the rings RES ] (a E Y). If S is a strong

semilattice Y of semigroups Sa (a E Y), then RES] is a strong

supplementary sum of the RESa] (a E Y). The results of

Chapter 3 can thus be applied to RES]. In particular, we

obtain the following results.


Theorem 5.4. Let 7 be a strict, hereditary radical

property. Then U(R[S]) = {A a E RES ]I PA E M*(U(REG ]);
aEY



Proof. The result follows from Theorem 3.10 and

Theorem 5.1.







We assume for Theorems 5.5 through 5.7 that RES] is a

strong supplementary semilattice sum of the R[S a (a E Y).


Theorem 5.5. Let Y be an m.u. semilattice. Then R[S]

is i-semisimple if and only if REG ] is :r-semisimple and,

for A e RES P AP a= 0 implies A = 0 for all a E Y.


Proof. The result follows from Theorem 3.6 and Theorem

5.1.


Theorem 5.6. Let Y be a semilattice in which all
*a
principal ideals are finite. Then R[S] = ZE R[S ] and
aEY
U(R[S]) = 9 E {A*clA E RES I and P A P EM* (U(RG ]);A ,I )}.
aEY a X py a a

Proof. The result follows from Theorem 3.7, Corollary

3.8, and Theorem 5.1.


Theorem 5.7. If U = J, G or N then U(R[S]) = {x E R[S]I

P (x )Pa E M* (U(RCG ]);A ,I ) for all a E Y}.


Proof. The result follows from Theorems 3.11, 3.13,

3.15, and 5.1.

Consider the special case in which S is a semilattice Y

of groups G (a E Y). Then S is a strong semilattice Y of

groups Ga (a E Y) [2, Theorem 4.111, and hence R[S] is a strong

semilattice sum of group rings REG j (a E Y). For each

a E Y, G = M(G ;I ,A ;P ), where II I = IA I = 1 and P is

the trivial matrix with entry e the identity in G We

therefore drop the representation of the REG ] (a E Y) by

matrix rings in the following results which follow directly

from Theorems 5.5 to 5.7.







1. Let Y be an m.u. semilattice. Then RLS] is 7-semi-

simple if and only if REG ] is r-semisimple for all a E Y.

2. Let Y be a semilattice in which all principal ideals

are finite. Then RES] = 9 E REG *I and U(R[S]) =
cEY
E U(R[G C)
aEY

3. If U = J, G or N, then U(R[S]) = {x E R[S]Ixa E

U(R[G ]) for all a E Y}.

Let S1 and S2 be semigroups, and let j: S1 S2 be a

homomorphism. Let R be a ring. Then > determines a ring

homomorphism from RES1] into R[S2] defined by

n n
Z r.s. E ri(s.i ),
i=l i=l

where r. E R and s. E S1 for i = l,...,n. We also denote this

ring homomorphism by for convenience.

If S is the trivial semigroup, then R[S] is clearly

isomorphic (and hence we may assume equal to) R. For any

semigroup ring RES], the mapping p: RES] R defined by

n n
E r.s. Z r.
i=l x i=l

is the ring homomorphism determined by the mapping from S onto

the trivial semigroup. We call p the augmentation homomorphism

of R[S]. The kernel of p, called the augmentation ideal of

RES] and denoted by w(R[S]), consists of all elements E r.s.
n i=l
of R[S] such that E r. = 0.
i=l 1







We next prove two results which will be used in the

determination of the 7-radical of any semigroup ring over a

completely 0-simple semigroup.


Lemma 5.8. If S is a semigroup with zero z, then

RLS] = R[z] 5 w(RLS]).


Proof. Let s E S and rlr2 E R. Then (rls)(r2z) =

rlr2sz = rlr2z and (r2z)(rls) = r2rlzs = r2rlz. Hence R[z]

is an ideal of R[S]. Clearly, R[z] n w(R[S]) = {0}. Let

x = Z rss e R[S]. Then x = ( r r )z + C[ r s ( E r )z] E
seS seS s.z s.z

R[z] + w(R[S]). Hence, RES] = REz] + w(RES]).


Lemma 5.9. Let S be a completely 0-simple semigroup

represented as a Rees matrix semigroup S = M(Go;I,A;P). Then

R[S] R 9 M* (R[G];I,A;P).
PY

Proof. By Lemma 5.8, R[S] = REz] I w(R[S]), where z is

the zero of S. Clearly, R[z] is isomorphic to R. Hence,

w(R[S]) = RES]/R[z] = M* (R[G];I,A;P) see [11, Lemma 3.11.
PY
We may assume that R[S] = R 9 M* (RLG];I,A;P).
PY

Theorem 5.10. Let S be a completely 0-simple semigroup

represented as a Rees matrix semigroup S = M(G ;I,A;P). Then

U(R[S]) = U(R) 9 {A E M* (R[G];I,A;P)IPAP E M*(U(R[G]);A,I)}.
PP Y

Proof. The result follows from Theorem 5.1 and Lemma


5.9.







Corollary 5.11. Let S be a completely 0-simple semigroup

represented as a Rees matrix semigroup S = M(Go;I,A;P). Then

RES] is i-semisimple if and only if R and REG] are fT-semisimple

and for A E M* (R[G];I,A;P), PAP = 0 implies A = 0.
PY

Proof. The result follows from Theorem 5.10 and

Corollary 4.6.

Results analogous to Theorems 5.4 through 5.7 can be

obtained for a supplementary semilattice sum of semigroup

rings over completely 0-simple semigroups.

Let Y be any semilattice, and let R be a ring. Clearly,

the semigroup ring R[Yj is isomorphicto (and hence we may assume

equal to) the strong supplementary semilattice sum of copies

of R indexed over Y with associated family {la 8: R RIa,8 E

Y, a B}. The following result follows from Theorem 3.19.


Lemma 5.12. U(REY]) = U(R)[Y] for any radical property.

Let S be a semilattice Y of semigroups S (a E Y). Then

the canonical semigroup homomorphism from S onto Y defines a

ring homomorphism n from R[S] onto REY] by


Z ( r s) s Z ( E rs ).
a(Y SES asY seS
a a

The kernel of n is Z w(R[S ]).
aY a


Theorem 5.13. Let S be a semilattice Y of semigroups

S (a E Y). Then U(RLES) s Z { Z r s ZE rs E U(R)}.
a a

Equality holds if and only if Z w(R[S ]) is a I-ring.
aEY a








Proof. Since U(REY]) = U(R)[Y] by Lemma 5.12, then
-1 -1
U(R[S3) s (U(REYI))n- = (U(R)EYI)nI = { Z r ss E rs
aEY SES seS
-1
E U(R)}. Furthermore, U(REY]) = (U(REY]))n1 /( E w(RES 3)).
aeY
The result follows from conditions (ii) and (iv).

The following result illustrates the usefulness of

Theorem 5.13.


Corollary 5.14. Let S be a strong semilattice Y of

rectangular bands Sa (a E Y). Then, if U = J, G or N,

U(R[S]) = E U(RS ) = { rs E r E U(R)}.
aXY aeY seS seS

Proof. For each a E Y, U(R[S ]) = { E r s l r E
SES sES
a a
U(R)} by the remarks following Corollary 5.2. Thus, for each

a E Y, w(R[S 3) S U(R[S ]). Hence, w(RSa ]) is a i-ring by

condition (ii) for each a E Y. By Corollaries 3.12, 3.14, and

3.16, E w(R[S ]) is a i-ring. The result follows from
aY a
Theorem 5.13.

In the case where Y is finite, we obtain a more general

consequence of Theorem 5.13. We first prove another lemma,

for which we do not need the full strength of hypotheses (i) -

(iv) on i.


Lemma 5.15. Let r be a hereditary radical property that

is extension closed. Let R be a supplementary semilattice sum

of rings R (a E Y) over a finite semilattice Y. Then R is a

T-ring if and only if Ra is a i-ring for all a E Y.







Proof. Suppose first that IYI = 2. Then Y = {a > 0}.

Hence, R/R = R and thus R is a i-ring if and only if R

and R are i-rings.

Assume the result holds for all semilattices of cardinality

< n (n 2 2). Let jYI = n. Let a E Y such that a is not the

zero of Y. Then F = R and I = E R are supplementary

semilattice sums over semilattices of cardinality < n. By

assumption, F and I are i-rings if and only if R is a

I-ring for all B E Y. Since R/I = F R is a Ir-ring if and

only if R is a i-ring for all 8 E Y. The result follows by

induction.


Corollary 5.16. Let S be a finite semilattice Y of

rectangular bands SC (a E Y). Then (RES]) = E U(R[S ]) =
aEY
r { Z r s Z r E U(R)}.
CeY seS sES
a a

Proof. For each a E Y, U(R[S ]) = { Z r s Z rs E U(R)}
SES seS
by Corollary 5.2. Thus, w(RLS 3) c U(R[S 1) for all a E Y.

Since n is hereditary, w(R[S ]) is a i-ring for all a e Y.

Since Y is finite, Z a(R[S 3) is a i-ring by Lemma 5.15. The
aeY
result follows from Theorem 5.13.

We note that any band is a semilattice of rectangular

bands. Corollary 5.16 thus determines the i-radical of any

semigroup ring over a band that is a finite semilattice of

rectangular bands.

We now consider the possibility of extending the pre-

ceeding results to the case of twisted semigroup rings.







Let p: S1 S2 be a semigroup homomorphism, and let R

be a ring with identity. Let yl: S, x S,1 U and Y2: S2 x S2

- U be twist functions into the group U of central units of

R. Define the mapping (: R t[Sl -+ R t[S2 by (rs)T = r(s-p)

for r E R, s SI. Clearly, preserves addition. For

s,s' E S1, (s*s')4 = (y1(s,s')ss')c = Y1(s,s')(ss =

Yl(s,s')(s ) (s'f), and (s$)-(s'4) = (s-) (s'#) =

Y2(s,s'#) (so) (s'(). Hence, 0 preserves multiplication if
and only if yl(s,s') = y2(so,s'#) for all s,s' E Sl. We call

j compatible if such an equation holds.

Let S be a strong semilattice Y of semigroups S (a E Y)

with associated family {(1 B: Sa Spja,8 E Y, a 6}. Let y

be a twist function from S x S into U. Let y denote the

restriction of y to S x S for a E Y. In general, c need

not be compatible with ye and y ; i.e., the equation

y(sat a) = Ya(s t a) = Y (s aa,,ta ,a)^ = Y(sa a,,tS aa,f)

need not hold for all s ,t E S and a,B e Y with a 2 B.

If iaB is compatible with ye and y for all a,8 E Y such that

a > 8, then Rt[SJ is a strong supplementary semilattice sum

of the twisted semigroup rings Rt [S (a E Y), and the results

of Chapter 2 and 3 are applicable.













REFERENCES


1. M. Aigner, Combinatorial Theory, Springer-Verlag, Berlin,
1942.

2. A. H. Clifford and G. B. Preston, The Algebraic Theory
of Semigroups, Vol. I, Vol. II, Amer. Math. Soc.,
Providence, R.I., 1961.

3. N. J. Divinsky, Rings and Radicals, Mathematical
Expositions, No. 14, Univ. of Toronto Press, Toronto,
1965.

4. T. W. Hungerford, Algebra, Holt, Rinehart and Winston,
Inc., New York, 1974.

5. B. J. Gardner, Radicals of Supplementary Semilattice
Sums of Associative Rings, Pacific Journal of Mathematics,
Vol. 58, No. 2, 1975, 387-392.

6. E. M. Patterson, On the Radicals of Certain Rings of
Infinite Matrices, Proc. Roy. Soc. Edinburgh Sect. A.
65, 1961, 263-271.

7. A. Quesada Rettschlag, Properties of Twisted Semigroup
Rings, Thesis, University of Florida, 1978.

8. M. L. Teply, E. G. Turman, and A. Quesada, On Semisimple
Semigroup Rings, Proc. Amer. Math. Soc., Vol. 79, No. 2,
1980, 157-163.

9. M. L. Teply, On Semisimple Semilattice Sums of Rings, to
appear.

10. J. Weissglass, Radicals of Semigroup Rings, Glasgow
Math. J., Vol. 10, 1969, 85-93.

11. J. Weissglass, Semigroup Rings and Semilattice Sums of
Rings, Proceedings of the American Mathematical Society,
Vol. 39, No. 3, 1973, 471-478.












BIOGRAPHICAL SKETCH

Eleanor Geis Turman was born January 18, 1948, in

Providence, Rhode Island. Her parents are James Brashears

Geis and Radm Lawrence Raymond Geis. In June, 1965, she

graduated from Wakefield High School in Arlington, Virginia.

In April, 1974, she received the degree of Associate of Arts

from Florida Junior College, Jacksonville, Florida. In

December, 1975, she received the degree of Bachelor of Arts

with a major in mathematics from the University of Florida,

Gainesville, Florida. She enrolled in the Graduate School

of the University of Florida in January, 1976, and in

June, 1977, she received the degree of Master of Science

with a major in mathematics. She held a Florida Board of

Regents Fellowship from January to June, 1976. She worked

as a graduate assistant in the Department of Mathematics

in the academic years 1976-1980. She was admitted to

candidacy for a Ph.D. with a major in mathematics in July, 1978.

Eleanor Geis Turman is the mother of a daughter,

Elizabeth Anne Turman.







I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


Mark L. Teply, Chairma
Associate Professor df Mathematics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of doctor of Philosophy.


Neil White
Associate Professor of Mathematics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


Alexander Bednarek
Professor of Mathematics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


David Drake
Professor of Mathematics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of philosophy.


Gerhard Ritter
/"Associate Professor of Mathematics
A-
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree ,f Doctor of Philosophy.


David Wilson
Associate Professor of Mathematics






I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in sco e and quality,
as a dissertation for the degree of D ct of Philosophy.


Ca s F.' Hoope r
Professor of Phsics

This dissertation was submitted to the Graduate Faculty of
The Department of Mathematics in the College of Liberal Arts
and Sciences and to the Graduate Council, and was accepted
as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.

August 1980


Dean, Graduate School












































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