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
UNIVERSITY OF FLORIDA
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