a generalization of fpf rings
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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
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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.
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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)
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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
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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) .
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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
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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
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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 -
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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
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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
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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.
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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.
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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 .
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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
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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.
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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
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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
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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
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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
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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.
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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
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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
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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
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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,
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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 ,
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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.
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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
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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
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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.
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884 BIRKENMEIER
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