This article was downloaded by: [University of South Florida]On: 02 May 2013, At: 00:14Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

A generalization of FPF ringsGary F. Birkenmeier aa Department of Mathematics, University of Southwestern Louisiana, Lafayette, LA,70504Published online: 27 Jun 2007.

To cite this article: Gary F. Birkenmeier (1989): A generalization of FPF rings, Communications in Algebra, 17:4,855-884

To link to this article: http://dx.doi.org/10.1080/00927878908823764

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial orsystematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution inany form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that thecontents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drugdoses should be independently verified with primary sources. The publisher shall not be liable for anyloss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arisingdirectly or indirectly in connection with or arising out of the use of this material.

COMMUNICATIONS IN ALGEBRA, 1 7 ( 4 ) , 835-884 ( 1 9 8 9 )

A GENERALIZATION OF F P F RINGS

Gary F. Birkenmeier

Department of Mathematics University of Southwestern Louisiana

Lafayette, LA 70504

Dedicated to the memory of Professor E. H. Fe l l e r

Lntroduction

It i s easi ly seen that if an R-module i s a generator in

the category mod-R, then it i s faithful. However, the con-

ve r se i s not necessar i ly t rue . Naturally one is led to a s k for

which rings i s i t t r u e that a faithful module is a generator.

These rings were introduced in 1966 and a r e called PF (pesudo-

Frobenius) rings. They were characterized by Azurnaya [lJ,

Osofsky [g, and Utumi [35]. In 1967 Endo [a initiated the

study of rings fo r which every finitely generated faithful

module i s a generator. These r ings a r e called FPF (finitely

pseudo-Frobenius) and were investigated by C. Faith, S. Page,

W. Burgess, S. Kobayashi, T. Faticoni, and H. Tachikawa.

Copyright O 1989 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

BIRKENMEIER

The purpose of this paper i s t o introduce a nontr ivial

general izat ion of the c l a s s of right FPF r ings . A ring R i s

said to be genera ted by faithful r ight cyc l ics , denoted r ight

GFC, if eve ry faithful cyclic right R-module genera tes the

category mod-R. Thus every r ight F P F , P F , o r QF (quasi-

Frobenius) r ing i s right GFC. The c l a s s of r ight GFC r ings

a lso includes commutat ive r ings (thus eve ry r ing h a s a GFC

subr ing-- i t s cen te r ) , s t rongly r egu la r r ings , and r ight con-

tinuous regular r ings of bounded index. Recal l a r ing R i s

(quasi-) Bae r if the r ight annihilator of every ( ideal) nonempty

subse t of R i s generated by an idempotent [g and [a. The

c l a s s of s emip r ime quas i -Baer right GFC r ings proper ly

contains the c l a s s of s emip r ime r ight FPF rings. F o r

example, commutat ive domains and strongly regular continuous

r ings a r e s e m i p r i m e quas i -Baer r ight GFC, but not neces -

s a r i l y r ight FPF. Our main r e su l t s a r e : (1 ) Let R be a

quas i -Baer r ight GFC ring; then ( i ) R i s s emip r ime iff R

i s right non-singular; ( i i ) .if R is semip r ime , then Q(R)

(i. e. , maxima l right quotient r ing of R ) i s FPF and

R = S 8 V, where S i s a product of p r i m e r ight Goldie

r ight GFC r ings and V i s a r ight GFC ring with z e r o r ight

soc le such that eve ry p r i m e idea l i s e s sen t i a l in V and Q(V)

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 857

h a s z e r o r ight socle. ( 2 ) Let R be a regular r ight se l f -

injective r ing; then R i s r ight GFC iff R i s a FPF ring

iff R has bounded index iff R i s left GFC. (3 ) Let R be a

regular ring. Then H i s a Baer r ight GFC ring iff R i s a

continuous ring of bounded index. (4) Semipr ime r ight G F C

r ings which a r e quas i -Baer o r quasi-continuous a r e

charac ter ized .

All r ings a r e assoc ia t ive , R will denote a ring with

unity, and Q(R) o r Q (when not ambiguous) will denote the

maximal right quotient ring of R . If X i s a s e t , then ;(X),

1 (X) , will symbolize the right, respect ively lef t , annihi lator - of X. Modules will be unital r ight R-modules. F o r a module

M, Zr(M) and t r a c e (M) will be the r ight s ingular sub-

module of M and t r a c e ideal of M, respect ively. Two-sided

concepts which a r e unmodified by "right" o r "left" mean both

s ides a r e sa t i s f ied (e. g . , X i s a n ideal means X i s a two-

s ided ideal , o r R i s FPF means R i s a left and r ight FPF

ring). A r ing d i r e c t summand i s a d i r ec t summand which i s

genera ted by a cen t r a l idempotent. Let X and Y be right

ideals such that X 5 Y 5 R. We say that X i s ideal essent ia l

in Y if eve ry nonzero ideal of R which i s contained in Y

h a s nonzero in tersec t ion with X. A right ideal i s reduced

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

858 BIRKENMEIER

if i t contains no nonzero nilpotent e lements . Other t e r m i -

nology can be found in [g, [14], [19], and [20]. Final ly, we

note that if R i s right GFC and X is a r ight ideal of R

such that R / X i s faithful, then Horn (R/X, X ) # 0.

This paper was inspired by and owes much to the

work of C. Fa i th and S. Page in the theory of FPF rings.

Thei r r e su l t s which a r e related to a proposition of this

paper will be indicated a t the beginning of the proposition

statement .

1. P re l imina r i e s ---

The f i r s t l emma provides a useful " internal" cha rac -

ter izat ion of a r ight GFC ring.

LEMMA 1. 1. The following conditions a r e

equivalent:

( i ) R i s right GFC.

( i i ) If M = mR is a faithful cycl ic module, then

l ( r ( m ) ) R = R. --

( i i i ) F o r every right ideal X of R such that X

contains no nonzero ideal , P(X)R = R (thus e ( X )

i s a genera tor in the ca tegory of lef t R-modules) .

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 859

( iv) F o r every right ideal X of R such that X con-

tains no nonzero ideal, R/X i s a generator.

PROOF. The resul t follows f r o m the fact that if X i s

a right ideal, then t r a c e (RIX) = P(X)R [L, p. 113, Ex. 81,

the fact that a module M is a generator if and only if

t r a c e (M) = R [L, p. 11 11, and the fact that R/X i s faithful

if and only if X contains no nonzero ideals.

A ring is right bounded if every essential right ideal

contains a nonzero ideal; i t i s s trongly right bounded if every

nonzero right ideal contains an ideal.

PROPOSITION 1. 2. (i) [c, Lemma C] Any rigkt

GFC ring i s right bounded.

( i i ) If R i s a strongly right bounded ring, then every

faithful cyclic module i s isomorphic to R; hence

R i s right GFC.

( i i i ) If R i s right GFC, then exactly one of the follow-

ing conditions hold: R is s imple Artinian; o r eve ry

maximal right ideal of R contains a nonzero ideal.

PROOF. (i) This proof follows the format of [g, Lemma

el. P a r t ( i i ) follows f r o m Lemma 1. 1. F o r par t ( i i i ) ,

le t M be a maximal r ight ideal such that M contains no

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

860 BIRKENMEIER

nonzero ideals . Then R has a cyclic faithful s imple module

RIM. Since t r a c e (RIM) = R , then R i s s imple Art inian.

COROLLARY 1. 3 . ( i ) [9, T h e o r e m s 1 & 21 If R i s -

e i ther selfbasic semiperfec t (e . g. , local), o r

reduced, o r right uniform, o r left duo, then R i s

r ight GFC if and only if R i s s trongly r ight

bounded.

( i i ) [u A ring i s r ight pr imi t ive r ight GFC if and

only if it i s s imple Art inian.

PROOF. ( i ) The selfbasic semiperfec t c a s e i s a

consequence of [s, Corol la ry 1. 2C] and Lemma 1, 1. The

remaining c a s e s follow f r o m Lemma 1. 1 and Proposi t ion 1. 2

s ince these r ings sa t i s fy the condition that whenever a nonzero

right ideal X contains no nonzero idea ls , then 1(X) c I # R - - where I i s an ideal of R. P a r t ( i i ) follows f r o m

Proposi t ion 1. 2 ( i i i ) .

Proposi t ion 1. 2 shows that the c l a s s of r ight G F C

r ings includes the c l a s s of duo r ings [18] ( e . g . , s t rongly

regular r ings) and thus the c l a s s of commutat ive r ings.

F r o m Corol la ry 1. 3 we s e e that the s t rongly r ight bounded

r ings f o r m a n important proper subclass of the c l a s s of

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 861

right GFC rings. Since right GFC rings satisfy the hypothesis

of the next proposition, it further indicates the relation between

right GFC rings and strongly right bounded rings. Also

par t ( i i i) of the following result generalizes [ 23 , Proposition 11 -

and [$ Proposition 2. 21.

PROPOSITION 1.4. Let R be a ring and X a nonzero

right ideal of R such that whenever R/X i s faithful, then

Hom (R/X,X) f 0.

(i) [s, Theorem 11 If Z ( R ) = 0, then R i s semiprime. r

( i i ) If Y i s a nonzero right ideal such that e (Y) n Y = 0,

then Y contains a nonzero ideal of R. Fur ther -

more, if Z (R) = 0, then Y is an essential r

extension of an ideal of R.

( i i i) Lf Z ( R ) = 0 and Y i s an essential right ideal r

of R, then Y is an essential extension of an ideal.

(iv) If Y i s a nonzero reduced right ideal of R , then

Y i s an essential extension of an ideal.

2 PROOF. (i) Assume A i s an ideal with A = 0. Let

B be a complement of :(A). If B = 0, then A c - Z ( R ) = 0. r

If B # 0, let H be an ideal in B; then AH 5 B n - r (A) = 0.

Hence H c - B 0 r (A) = 0. Let f : R/B -- B be a nonzero -

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

862 BIRKENMEIER

homomorphism such that f (1 t B) = b. Now ~ [ L ( A ) ] = 0.

Hence b(z(A) @ B) = 0 . Consequently, b a Z r ( R ) = 0.

Contradiction! Thus R i s s emip r ime .

( i i ) Suppose Y contains no nonzero ideal. Then R/Y

i s faithful. Hence, t he re i s a nonzero homomorphism

f : R/Y -- Y. Let f ( 1 t Y) = y; then y c P(Y) fl Y = 0.

Contradict ion! Thus Y contains a nonzero ideal. Le t I be

the s u m of a l l ideals of R contained in Y and K be a r e l a -

t ive complement of I in Y. Hence K contains no nonzero

ideal. Assume K # 0 . By the above a rgumen t , t h e r e ex is t s

0 # k E l ( K ) n K. Thus k(K 8 1) = 0. Since K 8 I i s

essent ia l in Y and Z (R) = 0 , then k c l ( Y ) n Y = 0. r

Contradict ion! Therefore , I i s essent ia l in Y. P a r t s ( i i i )

and ( iv) follow f r o m pa r t ( i i ) .

The proof of the following r e su l t i s s t raightforward.

PROPOSITION 1. 5. Let R = II R. be a finite product 1

of r ings Ri. Then R i s right GFC if and only if each R i i s

r ight GFC.

2. Semipr ime Quas i -Baer Right GFC Rings

Quas i -Baer r ings were introduced in [g and fur ther

developed in [3J and [_331. In [ z , p . 1681, s e m i p r i m e right

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 863

FPF r ings a r e shown to be quas i -Baer . F r o m Proposi t ion 1 .4 ,

a Baer GFC ring i s s e m i p r i m e . T h u s any commuta t i v e Baer

ring ( e . g . , a commuta t i v e domain) i s a s e m i p r i m e quas i -Baer

r ight GFC r ing; h o w e v e r , i n [z, p. 1881 i t i s shown that a c o m -

mu ta t i v e domain i s F P F i f and only i f it i s Pr l i f er . By [?,

T h e o r e m 2. 11 and Propos i t ion 1 . 2, a s t rongly regular continuous

ring i s a Baer GFC ring. T h u s any s trongly regular continuous

ring wh ich i s not s e l f i n j ec t i v e ( e . g . , [E, Example 13. 81) i s

s e m i p r i m e quas i -Baer GFC but not FPF [2, Coro l l a ry 2. I ] ,

[m. T h u s the c l a s s o f s e m i p r i m e quas i -Baer r ight GFC r ings

e f f e c t i v e l y genera l i zes the c la s s o f s e m i p r i m e r ight FPF r ings .

However , t h e r e a r e s e m i p r i m e right GFC r ings wh ich a r e not

quas i - Baer ( e . g . , Boolean r ings wh ich a re not s e l f i n j ec t i v e

[34 , - pp. 7 9 , 249, and 2501).

L E M M A 2 . 1 . T h e following s ta tements a r e equivalent:

( i ) E v e r y ideal i s right essen t ia l i n a ring d i rec t summand .

( i i ) E v e r y r ight ideal i s ideal essen t ia l in a ring d i rec t

summand .

PROOF. ( i ) 3 ( i i ) . Le t X be a r ight ideal i n R .

T h e n t he re e x i s t s a cen tra l idempotent b such that RX i s

essen t ia l in bR. L e t I be a nonze ro ideal i n bR such that

X n I = 0. T h e r e ex i s t s a cen tra l idempoten t e such that

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

864 BIRKENMEIER

I i s essential in eR 5 bR. Then RX c_ ( 1 - e)bR. Since RX

i s essential in bR, eR = 0 . Contradiction! Therefore, X

i s ideal essential in bR. The proof of the converse is

similar.

W e say R satisfies the (ideal) intersection left

annihilator sum property, (IILAS) ILAS, if whenever X and

Y a r e ( ideals) right ideals such that X fI Y = 0, then

( l ( X ) - + - l ( Y ) = R ) i ( X ) R t P ( Y ) R = R. IIRAS and IRAS a r e

defined similarly. Right FPF rings have the ILAS property

[ ~ , p . 1681.

LEMMA 2 . 2 . Let R be semiprime. Then the

following conditions a r e equivalent:

( i ) R i s a quasi-Baer ring.

( i i ) Every ideal i s essential in a ring direct summand

of R .

( i i i ) Every ideal which is a closed right ideal i s a

direct summand of R .

( iv) R i s an IILAS ring.

(v) R i s an IIRAS ring.

(vi) For every ideal X of R, z ( X ) i s essential in a

direct summand of R.

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 865

PROOF. ( i ) ( i i ) Assume R i s quasi-Baer and let

X be an ideal. Then there exists a central idempotent e such

that r ( X ) = (1 - e)R and X c - eR. Let Y be a right ideal in

2 eR such that X n Y = 0. Then (XY) = 0. Hence XY = 0.

Consequently, Y c - eR r) (1 - e)R = 0. Therefore X is

essential in eR. Clearly ( i i ) a (iii). F o r ( i i i ) + (iv), let

X and Y be ideals such that X n Y = 0. Then z(Y) is a

relative complement for Y. By [x, Proposition 1.41, r ( Y )

i s a direct summand. Since R is semipr ime, I(Y) = cR

where c i s a central idempotent. Hence X cR c - - P ( Y ) , so

(1 - c)R c - - P(cR) 5 P(X). Thus R is IILAS. For ( iv) *(i),

l e t X be a n ideal. Since R i s semipr ime, X n - r (X) = 0.

Then R = P(X) t P ( r ( X ) ) = 2(X) t ~ ( L ( X ) ) . But

r ( X ) n r ( l ( X ) ) = 0. Consequently, r ( X ) is a ring di rec t sum- -

mand. Thus R i s quasi-Baer. P a r t (v)@ (iv) follows by

left-right symmetry. Clearly ( i i ) 3 (vi) and f rom

[3, Proposition 101, (vi) +(i).

LEMMA 2. 3. ( i ) If every ideal of R i s essential in a

d i rec t summand, then R is IILAS.

( i i ) If R is right G F C and every ideal is essential in a

ring direct summand, then R sat isf ies ILAS.

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

8 66 BIRKENMEIER

Proof. ( i ) Let X and Y be ideals such that

2 X n Y = 0. Then X i s essent ia l in eR where e = e . Hence

R ( l - e ) c _ j ( e R ) c -- l ( X ) . Now eR Y = 0. Thus

R ( l - e ) t eR c_P(X) + P ( Y ) . Consequently, R i s IILAS.

( i i ) Let X and Y be r ight ideals such that X n Y = 0.

By Lemma 2. 1 t h e r e exist cen t r a l idempotents b and c such

that X and Y a r e ideal essent ia l in bR and cR, r e spec -

tively. Consider X = ( 1 - c)X $ cX. Now cX contains no

nonzero ideals. F r o m Lemma 1. 1, P(cX)R = R. But

l ( c X ) c = P(X)c. Thus bcR = bc l (cX)R = P(X)bcR c l (X)R . - - --

Consequently, ( 1 - b)R $ bcR 5 P(X)R. Now

1 = 1 - b t ( 1 - c ) b + bc E (1 - b)R + ( 1 - c)R + bcR

c - - l ( X ) R t L(Y)R. Therefore , R = - l ( X ) R t P(Y)R.

THEOREM 2.4. Let R be a s e m i p r i m e r ight GFC

ring. Then the following conditions a r e equivalent:

(i) R i s quas i -Baer .

( i i ) Every ideal i s essent ia l in a r ing d i r ec t summand.

( i i i ) R sa t i s f ies ILAS.

( iv) R sa t i s f ies IILAS.

PROOF. T h e proof follows f r o m Lemma 2 . 2 and

Lemma 2 . 3 .

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 867

In genera l , a s emip r ime quas i -Bae r ring i s not ILAS

(e. g. , a domain which does not sat isfy the right O r e condition).

PROPOSITICN 2 . 5 [31, Theorem 11 Let R be a quas i - -

Baer r ight GFC ring. Then R i s s emip r ime if and only if

R i s r ight nonsingular.

PROOF. Suppose R i s semipr ime. Le t

0 # x E Zr(R). Then t h e r e ex is t s a cen t r a l idempotent b such

that r (xR) = (1 - b)R and xRc_bR. But xR i s f a i th fu l and -

cyclic in bR. Therefore , xR genera tes bR. Le t

f : xR - bR be a nonzero homomorphism with f (x) = y. But

then y E Zr(R). Hence t r a c e (xR) = bR c Z (R). Contradic- - r

t ion! The re fo re Zr (R) = 0. The converse follows f r o m

Proposi t ion 1.4.

PROPOSITION 2.6 . Let R be a s emip r ime quas i -

Bae r r ight GFC ring. Then for each right ideal X t h e r e

ex is t s a cen t r a l idempotent b such that:

(i) r ( X ) = ( 1 - b)R = L(RX) c P(X), X i s ideal - - essen t i a l in bR, and RX i s essent ia l in bR;

( i i ) T h e r e exis t s a cent ra l idempotent c c bR such

that (1 - c)X i s essent ia l in ( 1 - c )bR, R / c X i s

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

868 BIRKENMEIER

a generator, and P(X)R = cR $ (1 - b)R = ( 1 - b + c)R;

( i i i) If X is principal, then t race (X) = bR.

PROOF. (i) The proof is routine.

( i i) Let Y be a complement of X in bR. Then there

exists a central idempotent e such that Y i s ideal essential

in eR. Let c = eb. Then Y i s ideal essential in cR and

X = (1 - c)X @ cX. Now ( 1 - c)X is essential in (1 - c)bR

and cX contains no nonzero ideals. By Lemma 1. 1, R/cX

i s a generator and j(cX)R = R. Thus - P(X)cR =P(cX)cR = cR.

Now - P(X)R =/(X)cR $ P(X)( l - c)bR (B - P(X)(l - c ) ( l - b)R.

Since (1 - b ) R c _ i ( X ) and (1 - b ) R s (1 - c)R, then

I ( X ) ( l - c ) ( l - b)R = (1 - b)R. By Proposition 2. 5 , (1 - c)bR -

i s a right nonsingular ring. Hence - P(X)(l - c)bR

= - 1 ( ( 1 - c)X)( l - c)bR = 0. Therefore, 4(X)R = cR d (1 - b)R

= (1 - b + c)R.

(iii) Since X is faithful and cyclic in bR, then X

generates bR. Therefore, t r ace (X) = bR.

COROLLARY 2 . 7 . Let R be a semipr ime quasi-Baer

right G F C ring.

(i) [E, Proposition 3. 261 Annihilator left ideals a r e

idempotent.

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 869

(") i - 11, Proposi t ion 4B] and [ 3 1, Theorem 31 R i s r ight -

co-nonsingular, hence R i s left nonsingular,

2 ( i i i ) Lf eR i s a minimal r ight ideal where e = e ,

then e i ther e i s cen t r a l o r ReR = bR is a s imple

Art inian ring which i s not a division ring where b

i s . a cen t r a l idempotent and R ( l - e)R = R.

( iv) Let X and Y be r ight ideals of R such that X

i s essent ia l in Y; then Y contains no nonzero

ideals if and only if X contains no nonzero ideals .

(v) If xR i s a n idempotent principal right ideal, then

t h e r e ex is t s a cen t r a l idempotent e such that

RxR = eR.

PROOF. (i) Let S c_ R, and le t SR denote the r ight

idea l genera ted by S. F r o m Proposi t ion 2.6 ( i i ) , t h e r e

ex is t s a cen t r a l idempotent d such that L(SR)R = dR. But

P(S) = f (SR). Thus 1 (S)P (S) = P(S)[RP(S)] = [P(S)R]P(S) - - - -

= [dR]P(S) - = dP(S) = L(S).

( i i ) Th i s p a r t follows f r o m Proposi t ion 2.6 ( i ) and ( i i )

and [3.

( i i i ) This p a r t follows f r o m Proposi t ion 2.6.

( iv) Suppose Y contains a nonzero ideal I and X

contains no nonzero ideals. By Proposi t ion 2.6 t h e r e ex is t s

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

870 BIRKENMEIER

a cen t r a l idempotent e such that I i s essent ia l in eR. Le t

X1 = X n eR and R = eR. Then X i s a n e s sen t i a l r ight ideal 1 1

in R F r o m Proposi t ion 2. 5, P(X1) = 0 in R But by 1' 1'

Lemma 1. 1, Q(X1)R1 = R1. Contradict ion! Therefore , Y

contains no nonzero ideals . The converse i s obvious.

(v) L e t x r R. By Proposi t ion 2 . 6 t h e r e ex is t s a

cent ra l idempotent b such that t r a c e (xR) = bR. Le t

f : xR -- R be a nonzero homomorphism such that f (x ) = y.

Then f(x)R = f(x)RxR = yRxT c_ RxR. The re fo re ,

t r a c e (xR) = RxR = bR.

THEOREM 2.8. [12, p. 158 Theorem C] and [31, - Theorem 21 Let R be a s emi -p r ime quas i -Bae r r ight GFC

ring. Then Q(R) is right GFC.

PROOF. Let M be a faithful cycl ic Q-module. The

singular submodule of M, a s a Q-module, i s the s a m e a s the

s ingular submodule of M a s a R-module. Suppose M/Zr (M)

i s not faithful a s a Q-module; then M / Z (M) i s not faithful r

a s a R-module. Let A = - r ( M / Z r ( M ) ) 5 R. Note Zr(M) # 0

and A f 0. Since A i s a n ideal, Theorem 2 .4 indicates that

t h e r e ex is t s a cen t r a l idempotent e r R such that A i s

essent ia l in eR. Le t m r M and cons ider

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS

me(A $ (1 - e )R) = meA = rnA 5 Z (M). Hence r

m e + Zr(M) e Z r ( M / Z r ( M ) ) = 0 since R i s right nonsingular

by Proposi t ion 2. 5. Thus Me 5 Zr(M). Consequently,

eR = A . Le t R = eR, e Q = Q1, and M = M e . Now 1 1

M I = Zr(M1). T h e r e exists a n epimorphism g : Q1 - M1.

Let h : R1 - Q1 be the inclusion. Let N = gh(R ), and 1

le t A = r (N) in R If A = 0, then N i s a genera tor 1 - 1' 1

and singular . Contradiction! Thus A is a nonzero ideal

of R1. Let 0 # a e A1. Then a Q l = cQ1 where

2 c = c E Q1. Hence c a = a , s o aR = caRIC_cR1. By 1

Proposi t ion 2.6 t h e r e ex is t s a cen t r a l idempotent b e R

such that aR1 5 bR and 2(aR ) = 0 in bR Le t R = bR1 1 1 1' 2

and Q2 = bQ1. Thus aR1 = caR 5 cR2. Hence ~ ( c R ) = 0 1 2

in R Consequently, t r a c e (cR ) = R Le t f : cR2 --R 2' 2 2' 2

be a nonzero homomorphism such that f ( c ) = y. Since m a p s

extend t o Q2, yc = y. But c = a q where q E Q Thus 1'

Ny = Nyc = Nyaq = 0, s o y E A1. Heilce t r a c e (cR ) = R 2 2

= b~~ 5 ~ ~ . Let H = {x E M I I xbQl = 0) # M1 since M i s

faithful. Now N 5 H, hence g ( e ) e H. Thus H = M 1'

Con t rad ic t~on! Therefore , M / Z r ( M ) i s faithful.

Now we only need t o cons tder nonsingular faithful

cyclic Q-modules. Let M be such a module. T h e r e exis t s

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

87 2 BIRKENMEIER

a n epimorphism g : Q - M. Let h : R -- Q be the inclusion

Le t N = gh(R) and let A = r ( N ) 5 R. Suppose A { 0. By

Theorem 2 . 4 , t h e r e e x ~ s t s a cent ra l idempotent e such that

A i s essent ia l in eR. But de (A @ ( 1 - e )R) = NeA = 0.

Hence Ne 5 Zr(M) = 0. Thus eR = A, Let

H = { x E M I xeQ = 0) f M since M i s faithful. Then N 5 H,

and H i s a submodule of M s ince eQ is an idea l of Q.

Hence g(1) E H. Thus H = M. Contradiction! The re fo re ,

A = 0. Hence N genera tes R. Therefore , t h e r e ex is t s a n

k epimorphism f : @ Ni - R for some positive integer k

i = 1

where N - N. Since maps extend t o Q, t h e r e ex is t s a homo- i -

k k morph i sm f l : @ Mi - Q such that 1 e f ( 8 Mi) where

i = 1 1 i = l

Mi - M. Hence f i s an epimorphism. Therefore , M 1

genera tes Q.

3. Regular Selfinjective G F C Rings

In this section we will cha rac t e r i ze those regular r ight

selfinjective r ings which a r e GFC. Using th is resu l t we

obtain a decomposition for a s emip r ime quas i -Baer right G F C

ring and a charac ter iza t ion of regular Bae r right G F C rings.

We note that the c l a s s of regular r ight self inject ive right G F C

r ings ( in fact , the c l a s s of r ight nonsingular r lght CS r ight

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS

G F C rings [z, Theorem 2. 11) i s contained in the c l a s s of

s emip r ime quas i -Bae r right G F C rings.

2 A ring i s cal led r ight fully idempotent if X = X for

eve ry r ight idea l X. A ring i s cal led biregular if eve ry

principal ideal i s genera ted by a cen t r a l idempotent. We note

that regular and b i regular r ings a r e r ight fully idempotent.

The following r e su l t i s a d i rec t consequence of

Corol la ry 2.7 (v).

PROPOSITION 3. 1. [32 , - Proposi t ion 51 Let R

be a quas i -Baer right GFC ring. Then R i s r ight fully

idempotent if and only if R i s biregular .

Recal l that R i s d i rec t ly finite if whenever xy = 1,

then yx = 1.

LEMMA 3. 2. Le t R be r ight nonsingular ring.

Then R contains no infinite d i rec t s u m s of nonzero p a i r -

wise isomorphic right ideals if and only if Q(R) i s direct ly

finite.

PROOF. The proof follows f r o m [14, - Proposi t ion 5. 91 and

[13, Proposi t ion 5.71.

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

874 BIRKENMEIER

PROPOSITION 3 . 3 . [ 2 9 , - Proposi t ion 71, [30] Let R

be a s emip r ime quas i -Baer r ight G F C ring. Then R con-

ta ins no infinite d i r ec t sums of nonzero pa i rwise isomorphic

r ight ideals; hence R and Q(R) a r e d i rec t ly finite.

PROOF. F r o m Theorem 2.8, Q(R) i s a r egu la r r ight

selflnjective right G F C ring. The proof of 129, Proposition 71 shows -

that Q(R) i s d i rec t ly finite. Application of L e m m a 3. 2

completes the proof.

See [g, pp. 110-1271 fo r the theory of "types" of a

regular r ight selfinjective ring.

LEMMA 3 . 4 . [z, Proposition 81, [g Let R be a regular

r ight selfinjective r ight GFC ring of type 11. Then R = (0) .

PROOF. Assume R # 0. By Proposi t ion 3 . 3 , R i s

d i rec t ly finite. Since type I1 r ings cannot be s imple Art inian,

Corol la ry 1 .3 indicates that R cannot be s imple. T h e r e -

fo re , t h e r e ex i s t s e = e 2 . R such that 0 # RelR # R. By 1 1

Corol la ry 2 . 7 (v), t h e r e ex is t s a cen t r a l idempotent b such

that RelR = bR. Let HI = (1 - b)R = f l (RelR) # 0. Now H1

and Re R a r e type I1 right GFC r ings s o we can r epea t the 1

p r o c e s s on RelR t o obtain H c Re R. Continuing in th is 2 - 1

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 87 5

way we obtain H 8 H2 $ . . . 5 R so that each H. i s a non- 1

zero directly finite ring direct summand of R . F r o m [g,

Proposition 10. 281, there exists an idempotent f . e Hi such

i that H.=Q fi R and fi.R _- f.R for a l l j 5 i. By

1 j=1 j J Proposition 2.6 the re exists a centra l idempotent c such that

00 O Hi is essential in cR. F o r m IIiZl fiR and let M be the

cyclic submodule ( ( f . ) . )R1, where R = cR and 1 1r1 1

I = {1,2 ,... ). Suppose x c r ( M ) c R 1 . Then (f.R )x = 0 1 1

for a l l i c I. Hence (8 H.)x = 0. But 8 H. i s faithful in R 1 1 1'

Thus M is a cyclic faithful R -module; hence the re exists 1 n

a positive integer n and an epimorphism t : @k=l Mk + R1

where Mk M. There exists

n Y = [ ( ( f i ) i e I ) ~ l ' ( ( f i ) i s I ) ~ 2 > ' ' 9 ( ( f i ) i e I ) ~ n ] ' Mk such that

t (y ) = c. There exists a central idempotent h such that

n t l H n t l

= hR1 = ejZl f n t l Rl . Observe f.h = 0 for i p n t 1 j

1

and f h = f Therefore, h = t (y)h = t[fy , fy n t l n t l ' I 2 , . . S t fynI

where f = ( 0 , . . . , 0, fn t l , 0 , . . . , 0) = ((f ) )h. Hence the re i s i ieI

an epimorphism f rom f y R t o H @i=l n t l i 1

Since n t l '

f n t l ~ i R 1 5 fn+lR 1 and H i s injective, the re i s an epi-

n t l n

morphism from BiZl (fntlR l ) i to Hntl. Since H i s n t 1

projective, the re is a monomorphism from

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

876 BIRKENMEIER

di rec t ly finite. Contradiction! Therefore R = (0).

THEOREM 3 . 5. [&, p. 2191, [10], [x, Theorem 91, [30]

Let R be a regular r ight selfinjective ring. Then the following

conditions a r e equivalent:

( i ) R i s a r ight GFC ring.

( i i ) R i s a FPF ring.

( i i i ) R has bounded index of nilpotence.

( iv) R i s a left GFC ring.

PROOF. ( i ) ( i i ) . F r o m [ z , Theorem 10. 221,

Proposi t ion 3 .3 , and Lemma 3.4, it follows that R i s of type I f '

Thus R has a faithful abelian idempotent e [ z , p. 1111. Since

R i s r ight GFC, eR i s a genera tor . Hence, R i s Mori ta

equivalent to a s trongly regular ring. By [29, - Corol la r ies 9. 1

& 9.21 and [a, R i s FPF.

( i i ) e ( i i i ) See [29, - Theorem 91, [g.

( i i ) -(iv) Obvious.

( iv) + ( i i ) R i s a regular Baer left GFC ring. Let

QL(R) denote the maximal left quotient r ing of R . F r o m

right- lef t symmet ry of Theorem 2.8, QI(R) i s a r egu la r lef t

selfinjective left GFC ring. Thus Q (R) i s a FPF ring. By 1

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 877

[20, - Theorem 7. 181, Ql(R) = Q(R) = R . Therefore , p a r t ( i i )

i s satisfied.

COROLLARY 3 . 6 . Let R be a s emip r ime quas i -Bae r

r ight GFC ring. Then G(R) i s a FPF ring.

Since a regular r ight FPF ring is selfinjective, the

c l a s s of regular r ight FPF rings i s contained in the c l a s s of

regular Baer right G F C rings. The containment i s p rope r

because strongly regular continuous r ings a r e r egu la r Baer

r ight GFC r ings which a r e not neces sa r i l y r ight FPF.

The next t heo rem cha rac t e r i ze s the c l a s s of regular Baer

r ight GFC rings.

COROLLARY 3 . 7 . Let R be a regular ring. The

following conditions a r e equivalent:

(i) R i s a Baer right GFC ring.

( i i ) R i s a right continuous r ing of bounded index.

( i i i ) R i s a continuous ring of bounded index.

(iv) R i s a Baer GFC ring.

PROOF. (i) ( i i i ) Assume R i s a Bae r r ight GFC

ring. By Corol la ry 3 . 6 and Theorem 3 .5 , R has bounded

index. F r o m [20, - Theorem 7. 181 and [z, pp. 79 & 2521,

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

878 BIRKENMEIER

R i s continuous. Clear ly ( i i i ) =? ( i i ) and ( iv) + ( i ) . F o r

(ii) 4 ( iv) , & pp, 79 & 2521 indicate that R i s Baer.

F r o m @, Theorem 13. 171, R = A 8 B where A i s a s trongly

regular continuous r ing and B i s a r egu la r r ight selfinjective

r ing. By T h e o r e m 3. 5, B i s a FPF ring. The re fo re , R i s

a GFC ring.

In [37] var ious conditions a r e given which, when

coupled with Corol la ry 3.7, i n su re that a r egu la r Baer r ing

i s GFC o r that a regular right continuous r ing i s GFC.

COROLLARY 3.8. [z, Corol la ry 2A] A r ing R i s

a p r i m e regular r ight GFC ring if and only if R i s s imple

Artinian. Hence, a p r i m e r ight GFC ring i s a r ight Goldie

ring.

PROOF. Suppose R i s a p r i m e r egu la r r ight GFC

ring. By T h e o r e m 3. 5 and [z, Corol la ry 2A], Q(R) i s s imple

Artinian. It follows that R i s s imple Art inian. The converse

i s obvious. The second s ta tement follows f r o m the fac t tha t

a p r i m e r ing i s quas i -Baer , Corol la ry 3.6, and the f i r s t

statement.

PROPOSITIOI~ 3.9. [g, Theorem 8A] A r ing R i s a

p r i m e right GFC ring if and only if it i s r ight nonsingular ,

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS

r ight bounded, and L(X)R = R for every inessent ia l r ight

ideal X.

PROOF. Assume R i s a p r i m e r ight GFC ring.

Clearly, i t i s r ight nonsingular. By Proposi t ion 1. 2, R i s

r ight bounded. F r o m Lemma 1. 1 and the fact that a l l ideals

a r e essent ia l in R, then e(X)R = R for every inessent ia l

r ight ideal X. Conversely, by Lemma 1. 1 R i s r ight GFC.

Let X and Y be ideals such that XY = 0 and Y # 0. Then

Y m u s t be essential . Hence X = 0. Therefore , R i s

pr ime.

LEMMA 3. 10. Le t R be a r ing such that every ideal

which i s a closed r ight idea l i s a d i r ec t summand of R. If

R i s a subring of a r ing T such that R i s essent ia l in T

a s a n R-module, then every cen t r a l idempotent of T i s in R.

PROOF. Le t e be a cen t r a l idempotent of T . Le t

2 Hence, t he re ex i s t s c = c E R such that K = cR. But cT

i s essent ia l in eT. Thus cT = eT. Hence e = c E R.

THEOREM 3. 11. Let R be a s emip r ime quas i -Bae r

r ight GFC ring.

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

BIRKENMEIER

( i ) R = S $ V where S i s a product of p r i m e r ight

Goldie right GFC r ings and V is a r ight GFC ring with z e r o

r ight socle such that every p r ime ideal of V i s essent ia l in

V and Q(V) h a s z e r o right socle.

( i i ) [c, Theorem 1.41 If R contains no infinite s e t of

orthogonal cen t r a l idempotents, then R i s a finite product of

p r i m e r ight Goldie r ight GFC rings.

PROOF. F r o m Corollary 3. 6, Q(R) i s a r egu la r se l f -

injective FPF ring. By Lemma 2. 2 and [=, Theorem 9. 131

Q(R) = aQ(R) cB bQ(R) where a and b a r e cen t r a l idempotents

of Q(R) and aQ(R) i s a product of s imple Art inian r ings and

bQ(R) has z e r o right socle. L e m m a s 3. 10 and 2. 2 yield that

a , b r R and a Q ( R ) = Q ( S ) = n Q(Si) where aR = S = n S ir I ir I i

and each S. i s a p r i m e right Goldie r ight GFC ring. Now

bQ(R) = Q(V) i s a regular self injective FPF ring with z e r o

r ight soc le where V = bR. Le t X be a homogeneous component

of soc (V). By Corol la ry 2.7 ( i i i ) , X = cR where c i s a

cent ra l idempotent and cR is a s imple Art inian ring. Hence

X = cQ(R) c_ soc (Q(R)) 5 aQ(R). Thus soc (V) = 0. Lemma 2 . 2

( i i ) shows that every p r i m e ideal of V i s essent ia l .

( i i ) F r o m Lemma 3. 10, Q(R) has no infinite s e t s of

orthogonal cen t r a l idempotents. By Theorem 3. 5 and

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 881

Corol la ry 3 .6 , Q(R) i s a regular right selfinjective ring of

bounded index. F r o m [20, - Theorem 7. 20 and Corol la ry 6.81,

Q(R) i s semis imple Artinian. Applying Lemma 3. 10 yields

the resul t .

A r ing is right CS [5J if every r ight idea l i s essent ia l

in a d i rec t summand. F r o m [22), R i s right quas i -

continuous if i t i s a right CS r ing such that if X and Y a r e

d i r ec t summands of R with X n Y = 0, then X 8 Y i s a

d i r ec t summand of R. The following r e su l t genera l izes

Theorem 2 of [25 111. -

THEOREM 3 . 12. Let R be a s emip r ime r ight GFC

ring. Then the following conditions a r e equivalent.

( i ) R i s right quasi-continuous.

( i i ) R = A d B where A i s a reduced Bae r ring and

B i s isomorphic to a finite d i r ec t s u m of full m a t r i x r ings

over s trongly r egu la r selfinjective r ings.

( i i i ) R and Q(R) have the s a m e se t of idempotents.

PROOF. ( i ) ( i i ) By Lemma 2. 2 , Propos i t ion 2. 5,

and [i, Theorem 2. 11, R i s a Baer ring. F r o m

[A, Corol la ry 131, R = A 8 B where A i s a reduced Bae r

ring and B i s a DN ring. By [z, Corol la i re 4. 141, B i s a

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

882 BIRKENMEIER

right self inject ive regular ring. F r o m T h e o r e m 3. 5 and

[20, - Theorem 7. 201, i t follows that B i s i somorphic t o a

finite d i r ec t s u m of full m a t r i x r ings over s trongly regular

r ings .

( i i ) + (iii) Since A i s reduced, t h i s implication

follows f r o m Corol la ry 2.7 ( i i ) , [ , T h e o r e m 2. 11, and

Lemma 3. 10.

( i i i ) +(i) Th i s implication follows f r o m [ 2 7 ,

Proposi t ion 3. 11.

REFERENCES

1. G. Azumaya, Completely faithful modules and self - injective r ings , Nagoya J. Math. 27 (1966), 697-708.

2. F. W. Anderson and K. R. Fu l l e r , Rings and Categor ies of Modules, Spr inger-Ver lag , New York, N. Y . , 1974.

3. G. F. Birkenmeier , Baer r ings and quasi-continuous r ings have a MDSN, Pacif ic J. Math. 97 (1981), 283-292. -

4. W. D. Burgess , On nonsingular r ight FPF r ings , Comm. Algebra 12 ( l984) , 1729- 1750.

5. A . W. Chat te rs and S. M. Khuri , Endomorphism rings of - modules over non-singular CS r ings , J. London Math. SOC. 21 (1980), 434-444. -

6. W. E. Clark , Twisted m a t r i x units semigroup a lgeb ras , Duke Math J. - 34 (1967), 417-424.

7 . S. Endo, Completely faithful modules and quas i -Frobenius a lgeb ras , J. Math. Soc. Japan 2 (1967), 437-456.

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

A GENERALIZATION OF FPF RINGS 883

8. C. F a i t h , A l g e b r a 11: Ring T h e o r y , Spr inger -Ver lag , B e r l i n , 1976.

9 . , In jec t ive c o g e n e r a t o r r i n g s and a t h e o r e m of Tachikawa I, 11, P r o c . A m e r . Math. S o c . , 60 (1976), 25-30, 6 2 (1977) 15-18.

10. , C h a r a c t e r i z a t i o n s of Rings by Fa i th fu l Modules , L e c t u r e Notes , Math. Dept . Technion, Haifa, 1976.

11. , S e m i p e r f e c t P r u f e r and FPF r i n g s , I s r a e l Math. J. 26 (1977), 166-177. -

12. , In jec t ive quot ient r i n g s of commuta t ive r i n g s , in Module T h e o r y , S p r i n g e r L e c t u r e Notes 700, S p r i n g e r - V e r l a g , B e r l i n , 1979, 151-203.

13. h j e c t i v e Modules a n d I n j e c t ~ v e Quotient Rings , L e c t u r e Notes in P u r e and Applied Math. vol. 7 2 , M a r c e l Dekker , New York , 19d2.

14. C. Fa l th and S. P a g e , FPF Ring T h e o r y : Fai thful Modules a n d G e n e r a t o r s of Mod-R, London Math. Soc. L e c t u r e S o t e s S e r l e s 88, Cambridge L n l v e r s l t y P r e s s , Cambr tdge , 1984.

15. T . G. F a t ~ c o n l , FPF r i n g s I: T h e Noether ian c a s e , C o m m . A l g e b r a , 13 (1985), 2119-2136. -

16. , S e m 1 - p e r f e c t F P F r l n g s and a p ~ l ~ c a t l o n s , J . A l g e b r a , 107 ( 1 Q 8 7 ) , 207-315. -

17. - , Loca l iza t ion in f in i te d i m e n s ~ o n a l FPF r i n g s , t o a p p e a r .

18. E. H. F e l l e r , Properties of p r i m a r y n o n c o m m u t a t l ~ - e r l n g s , T r a n s . A m e r . Math. Soc. 89 (19581, 79-91.

19. K. R. Goodear l , R m g T h e o r y : K o n s ~ n g u l a r Rings a n d Modules , P u r e and A p p l ~ e d M a t h e m a t i c s S e r ~ e s 33, M a r c e l D e k k e r , New York, 1976.

20. , Von Neumann R e g u l a r Rings, P i t m a n , London, 1979.

21. K. R. G o o d e a r l a n d A. K. Boyle, Dimenston t h e o r y f o r n o n s i n g u l a r ~ n j e c t ~ v e m o d u l e s , M e m o i r s A m e r . Math. Soc. 177, 1976.

22. L. J e r e m y , Modules e t anneaux quas l -con t inuous , Canad. Math. Bull. 17 (1974), 217-228. P

23. H. K a m b a r a and S. Kobayashi , On r e g u l a r self-injective r i n g s , Osaka J. Math. 22 (1985), 7 1-79.

Dow

nloa

ded

by [

Uni

vers

ity o

f So

uth

Flor

ida]

at 0

0:14

02

May

201

3

884 BIRKENMEIER

24. I. Kaplansky, R ~ n g s of O p e r a t o r s , M a t h e m a t ~ c s L e c t u r e h o t e S e r i e s , W. A . Ben jamin , Nev. York , 1968.

2 5 . S. K o b a y a s h ~ , On non-s ingu la r F P F r l n y s I , I1 O s a k a J. l l a t h . 2 2 i 198 5), 787-795: 707-803. -

26. J. Lambek , L e c t u r e s on Rings a n d M o d u l e s . Blai s d e l l , Wal tham, h i a s s . , 1966.

27. K. O s h i r o , Cont inuous m o d u l e s arid q u a s i - c o n t i n u o u s modii les , O s a k a J . Math. LO 1 l 9 8 3 ) , 68 1-694.

28. B. L. Osofsky, A g e n e r a l ~ z a t i o n of q u a s l - F r o b e n i u s r i n g s , J'. ~A1cc5ra 4 119,7n\, 3-3-38:. - --

30. , C o r r e c t i o n to "Regula r FPF r i n g s , " P a c . J. M a t h . , 97 (1981) , 488-490. -

3 1. , S e m i - p r i m e a n d n o n - s i n g u l a r FPF r i n g s , C o m m . A l g e b r a , 10 (1982) , 2253-2259.

3 2. , S e m i h e r e d ~ t a r y and ful l idempoten t FPF r i n g s , C o m m . A l g e b r a , 11 (1983) , 227-242.

33. A . Po l l ingher a n d A . Zaks , On B a e r a n d q u a s r - B a e r r l n g s , Duke Math. J. 3; (1Q70) , 127-138.

34. B. S t e n s t r o m , Rings of Quot ien t s , G r u n d l e h r e n Math. W i s s . , Band 217, S p r i n g e r - V e r l a g , New York , 1975.

35. Y. Utumi, On r i n g s of which a n y o n e - s i d e d quo t ien t r i n g s a r e two- s ided , P r o c . A m e r . Math. Soc. 2 (1963) , 141-146.

36. , Sel f - in jec t ive r i n g s , J. A l g e b r a 6 (1967) , 56-64.

37. R. Yue Chi Ming, On r e g u l a r r i n g s a n d s e l f - i n j e c t i v e r i n g s , s. Math. 91 ( l 9 8 l ) , 152-166.

Rece ived : Sep tember 1987 R e v i s e d : J anua ry 1988 D

ownl

oade

d by

[U

nive

rsity

of

Sout

h Fl

orid

a] a

t 00:

14 0

2 M

ay 2

013