idempotents and completely semiprime ideals
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Idempotents and completely semiprime idealsGary F. Birkenmeir aa Department of Mathematics , Southeast Missouri State University , CapeGirardeau, Missouri, 63701Published online: 27 Jun 2007.
To cite this article: Gary F. Birkenmeir (1983) Idempotents and completely semiprime ideals, Communicationsin Algebra, 11:6, 567-580
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COMMUNICATIONS I N ALGEBRA, 11 (6 ) , 567-580 (1983)
IDEMPOTENTS AND COMPLETELY SEMIPRIME IDEALS
Gary F. Birkenmeier Department of Mathematics
Southeas t Missour i S t a t e Univers i ty Cape Girardeau, Missouri 63701
I n t h i s paper we show
{ei 1 i E 1 3 of a r i n g R:
n ( i . e . r < Y r E Y) then
(Theorem 4 ) t h a t f o r a s e t o f idempotents
( i ) i f Y i s a completely semiprime i d e a l
Y + C eiR i s a completely semiprime
i d e a l ; ( i i ) i f Y is a semiprime i d e a l such t h a t ( 1 - e i ) ~ e i Y
f o r a l l i E I and I is f i n i t e , t hen Y + e.R i s a semiprime i d e a l .
I n the main theorem (Theorem 13) we prove t h a t f o r a r i n g R t h e r e
e x i s t s a s e t of o r thogonal idempotents {e . 1 j E 53 such t h a t R J
i s an e s s e n t i a l ex tens ion of t h e completely semiprime i d e a l
S = (O e.R) G [ n ( 1 - e . )R] where n (1 - e.)R is a l s o a completely J J J
semiprime i d e a l which con t a in s every DN r i g h t i d e a l and G e .R J
i s maximal among reduced ( i . e . no nonzero n i l p o t e n t elements)
d i r e c t sums of idempotent genera ted r i g h t i d e a l s . This r e s u l t
gene ra l i z e s a r e s u l t of Utumi on von Neumann r e g u l a r r i ngs .
The s t r u c t u r e of a reduced r i n g has been s t ud i ed ex t ens ive ly .
One way t h i s s t r u c t u r e theory can be appl ied t o an a r b i t r a r y r i n g
i s by no t i ng t h a t t he qxmtient of a r i n g by a completely semiprime
i d e a l i s a reduced r i n g [IJ, [GI, [ G I . Another way t o apply t h i s
Copyright O 1983 by Marcel Dekker, Inc.
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568 BIRKENMEIER
theory would be t o decompose a r i n g i n t o a d i r e c t sum of a reduced
p a r t and a p a r t which c o n t a i n s t h e n i l p o t e n t e lements i n some
dense way [A], [k] , [A], [ L ] , [ s l y [KI. I n t h i s paper we
" e s s e n t i a l l y " ach ieve such a decomposi t ion (Theorem 13) by
i n v e s t i g a t i n g t h e i n t e r a c t i o n between idempotents and completely
semiprime i d e a l s .
Throughout t h i s paper a l l r i n g s a r e a s s o c i a t i v e : R denotes
a r i n g w i t h u n i t y ; N and N denote t h e s e t of n i l p o t e n t e lements n
and t h e s e t of n i l p o t e n t e lements of index n , r e s p e c t i v e l y . A
r i g h t ( l e f t ) R-module X i s a r i g h t ( l e f t ) e s s e n t i a l e x t e n s i o n of a
r i g h t ( l e f t ) R-submodule Y i f every nonzero r i g h t ( l e f t ) R-submodule
o f X has nonzero i n t e r s e c t i o n w i t h Y. A r i n g Q i s a r i g h t ( l e f t )
q u o t i e n t r i n g of i t s subr ing S i f f o r a l l x,y E Q w i t h x f 0
t h e r e e x i s t s s S such t h a t ys E S and x s # 0 ( s y E S and s x # 0).
The r i g h t a n n i h i l a t o r and t h e l e f t a n n i h i l a t o r of a s u b s e t V of R
a r e den0 t e d annr (V) and annL (V) , r e s p e c t i v e l y . For t h e b a s i c p r o p e r t i e s of a completely semiprime i d e a l s e e
L 1 , [a], 1l.3-I, and [El. I n [LI, [A], [&I, [?I, and [El completely - semiprime i d e a l s a r e c a l l e d semicompletely prime i d e a l s . The
g e n w a l i z e d n i l r a d i c a l N is a completely semiprime i d e a l which g
i s conta ined i n every completely semiprime i d e a l . I f N i s a n
i d e a l t h e n N = N Every completely semiprime i d e a l i s semiprime g'
and t h e c o n d i t i o n s a r e e q u i v a l e n t f o r r i g h t duo r i n g s ( i . e . every
r i g h t i d e a l i s an i d e a l ) [g]. However (von Neumann) r e g u l a r
r i n g s p rov ide examples of r i n g s where every i d e a l i s semiprime b u t
n o t n e c e s s a r i l y completely semiprime.
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COWLETELY SEMIPRIME IDEALS 569
A r i g h t i d e a l X i s dense ly n i l (DN) i f e i t h e r : X = 0; o r
X f 0 and every nonzero r i g h t i d e a l of R which is contained i n X
has nonzero i n t e r s e c t i o n w i th N 121. The no t i on of a dense ly n i l
r i g h t i d e a l gene ra l i z e s t h a t of a n e s s e n t i a l l y n i l p o t e n t r i g h t
i d e a l [ i ] and [%I, and determines t he c l a s s of r i g h t i d e a l s
which con t a in no reduced r i g h t i d e a l s o f R. The r i g h t s i ngu l a r
i d e a l Z p rovides an example of a DN i d e a l which i s no t necessar i -
l y n i l [A, Lemma 3.31.
This paper grew o u t of r e s u l t s on idempotents i n reduced
r i g h t i d e a l s , However i t became ev ident t h a t by de f i n ing t h e
no t i on of a r i g h t semicent ra l idempotent many of t h e r e s u l t s would
hold i n a more genera l contex t . An idempotent e i s r i g h t semi-
c e n t r a l i f eR = eRe ( equ iva l en t l y , f o r r € R then e r = e r e ) .
Hence every idempotent is r i g h t semicent ra l modulo a n i l p o t e n t
element f o r i f x = x2 then (x r - x rx )2 = 0. Since every idempotent
i n a reduced r i g h t i d e a l is r i g h t s emicen t r a l , one can s e e t h a t a
minimal r i g h t i d e a l i s e i t h e r DN o r genera ted by a r i g h t semi-
c e n t r a l idempotent. Also i f a n idempotent e of R i s r i g h t
s emicen t r a l and e i t h e r R i s semiprime o r e is i s l e f t semicent ra l
( i . e . Re = eRe), then e is c e n t r a l . From [A] and [GI t h e b a s i c
p r o p e r t i e s of a r i g h t s emicen t r a l idempotent a r e summarized i n
Lemma 1.
2 LEMMA 1. L e t e = e E R, then (i) through (v) a r e equiva len t :
( i ) e i s a r i g h t s emicen t r a l idempotent.
( i i ) ( 1 - e)R i s an i d e a l .
( i i i ) R( l - e ) = ( 1 - e)R(1 - e ) ( i . e . 1 - e i s l e f t s emicen t r a l ) .
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5 70 BIRKENMEIER
( i v ) ( 1 - e)Re is a n i d e a l and ( 1 - e ) ~ = ( 1 - e)Re 6 ~ ( 1 - e ) .
(v) R is r i n g isomorphic t o 1 eR 0
(1 - e)Re R ( l - e )
I
Furthermore i f e is a r i g h t s e m i c e n t r a l idempotent , t h e n
(a ) i f xL = x E R t h e n xR = A @ B where B 5 ( 1 - e)R and
(b) R/ (1 - e)R is r i n g isomorphic t o eRe = eR; every r i g h t
i d e a l of eRe is a r i g h t i d e a l of R, and every l e f t i d e a l
of ( 1 - e)R is a l e f t i d e a l of R.
One can s e e from p a r t s ii and iii t h a t i f every idempotent
i s r i g h t s e m i c e n t r a l o r i f every idempotent genera ted r i g h t i d e a l
i s a n i d e a l , t h e n every idempotent is c e n t r a l .
LEMMA 2. L e t {ei I i E 13 b e a s e t o f idempotent e lements
of R. I f Y is a n i d e a l such t h a t ( 1 - ei)Rei c Y + L eiR f o r a l l
i E I, then Y + C eiR is an i d e a l . I n p a r t i c u l a r i f N c Y + C eiR, 2 - then Y + 2 eiR is an i d e a l .
Proof . L e t r E R, then re i = eirei + ( 1 - e i ) re i E Y + 2 eiR.
Hence r(Y + C eiR) _c Y + C eiR. Therefore Y + 2 eiR is a n i d e a l .
LEMMA 3 . L e t {ei 1 i E I] b e a s e t of r i g h t s e m i c e n t r a l
idempotent e lements of R.
2 ( i ) ann ( C e .R) = n (1 - ei)R. Hence ( ( 2 eiR) n [ n ( l - ei)RI) =O. r 1
( i i ) I f I is a f i n i t e s e t , t h e r e e x i s t s a r i g h t semicen t ra l
idempotent e E L eiR such t h a t x e = x f o r a l l
x E C eiR and R = (C eiR) + [n ( 1 - ei)R].
( i i i ) I f l e i [ i E I ] i s an or thogona l s e t , then
S = (G eiR) G [n ( 1 - ei)R] i s a n i d e a l .
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COMPLETELY SEMIPRIME IDEALS
Proof . ( i ) . The proof i s r o u t i n e .
( i i ) . Suppose I = [l, 21. Then e = el + e 2 - e e w i l l be a 1 2
r i g h t s e m i c e n t r a l idempotent such t h a t x e = x f o r a l l x E e R + e R. 1 2
n+l e e ) + . + - 1 (ele2 . . . e ) where t h e s i g n between t h e
n-1 n n
p a r a n t h e s e s a l t e r n a t e s and t h e kth p a r e n t h e s i s is a sum o f p r o a u c t s
of e . w i t h each product having k f a c t o r s and t h e s u b s c r i p t s of t h e
f a c t o r s i n t h e p roduc ts form a l l combinat ions of n t h i n g s taken
k a t a t ime. S t r a i g h t f o r w a r d c a l c u l a t i o n w i l l show t h a t e h a s
t h e d e s i r e d p r o p e r t i e s .
Again suppose I = El, 23 and l e t r E R. Consider r = e r + 1
1 - e l But (1 - e ) r = e 2 ( 1 - e l ) r + ( 1 - e 2 ) ( l - e ) r E e R + 1 1 2
( 1 - e )R n ( 1 - e2)R by Lemma 1. Thus r E Z eiR + [n (1 - ei)R]. 1
The g e n e r a l c a s e fo l lows by i n d u c t i o n .
( i i i ) . L e t ( 1 - e ) x e E (1 - e )Rek and e j f ek where j , k E I. k k k
Then ( 1 - e ) x e = e . ( l - ek)xe k k J
k + ( 1 - e j ) ( l - ek)xek. But
e . ( l - ek)xek = 0 s i n c e e i s r i g h t s e m i c e n t r a l and e . e = 0. Thus J j J k
( 1 - e i ) R e i s n ( 1 - e . )R f o r a l l i E I. From Lemma 2, S i s an
i d e a l . Th is completes t h e p roof .
The r i n g of 2 x 2 upper t r i a n g u l a r m a t r i c e s over a reduced
r i n g R w i t h r i g h t s e m i c e n t r a l idempotents , O 0 and I 0 1 I
p rov ides an example f o r Lemmas 1, 2, and 3 . Note
i s a n i d e a l which c o n t a i n s N b u t i s n o t genera ted a s a r i g h t i d e a l
by a s i n g l e idempotent.
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572 BIRKENMEIER
The nex t theorem shows t h a t semiprime i d e a l s and completely
semiprime i d e a l s can b e en la rged by adding idempotent genera ted
r i g h t i d e a l s . We observe t h a t a s t r a i g h t f o r w a r d argument shows
t h a t a n i d e a l Y i s completely semiprime i f and o n l y i f whenever
r2 E Y then r E Y.
THEOREM 4. L e t le i 1 i E I ] be a s e t of idempotent e lements
of R.
( i )
( i i )
( i i i )
I f Y i s a completely semiprime i d e a l , t h e n Y + C eiR i s
a comple te ly semiprime i d e a l .
I f N c C eiR and ( 1 - ei)Rei i s a r i g h t i d e a l f o r a l l 3 -
i E I , then 2 e . R is a completely semiprime i d e a l .
Le t I b e a f i n i t e s e t and Y a semiprime i d e a l such t h a t
(1 - e . )Re g Y f o r a l l i E I. ThenY + LeiR i s a 1 i
semiprime i d e a l .
Proof . For a l l p a r t s l e t (el, e . . . , e ] b e a f i n i t e s u b s e t 2' n
of (ei 1 i E 13 and 1 + E = ( I - el) ( 1 - e2) . . . ( 1 - en) where E
is a sum whose terms a r e -e o r p roduc ts of -e f o r j E C1,2, . . . , n]. j j
2 i ) . L e t r E R such t h a t r E Y + C eiR. Then t h e r e e x i s t s
y E Y , {e . 1 j = 1, 2, ..., n], and x E R such t h a t J j
r2 = y + elxl + e x + . . . + e x S i n c e a completely semiprime 2 2 n n'
2 i d e a l c o n t a i n s N , ( 1 -ei)Re. c Y. Thus ( 1 + E)r E Y. From
1 - 1 2 , Lemma 1 1 , ( 1 + E ) r E Y. Hence r E Y + eiR. BY Lemma 2,
Y + C eiR i s a n i d e a l . Therefore Y + 2 e R is a completely i
semiprime i d e a l .
2 ( i i ) . L e t r E R such t h a t r E 1 e R. Then t h e r e e x i s t s i
L ( e j I j = 1, 2, . .. , n ] and x E R such t h a t r = e x + e x
j 1 1 2 2
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COMPLETELY SEMIPRIME IDEALS 573
2 + ... + e x . Observe ( 1 + E ) r E ( 1 - el)Rel + ( 1 - el) ( 1 - e2)Re2 n n
+ ... + ( 1 - e l ) ( l - e2) ... ( 1 - en-l)Ren-l s i n c e (1 - ei)Rei i s
a r i g h t i d e a l f o r a l l i E I. Therefore ( 1 + ~ ) r ' ( 1 + E) = 0,
By s i m i l a r c a l c u l a t i o n s i t fo l lows t h a t [ ( I + ~ ) r ] ~ = [ ( l + E ) r 2
+ ( 1 + E)rEr] ( 1 + E ) r = 0. Consequently, ( 1 + E ) r E C eiR. Hence
r E C eiRi. From Lemma 2, i t f o l l o w s t h a t Z e .R i s a completely
semiprime i d e a l .
( i i i ) . L e t a E R such t h a t aRa Y + C eiR and {ei ( i E I ] =
( e j I j = 1, 2 , ..., n]. Then f o r each r E R t h e r e e x i s t s y t Y
and x E R such t h a t a r a = y + elxl + e x + ... + enxn. Now j 2 2
( 1 + E)ara E Y because ( 1 - ei)Rei _C Y f o r a l l i E I. Thus
( 1 + E)aR(l + E)a _c Y . Hence ( 1 + E)a E Y s i n c e Y i s a semiprime
i d e a l [?,p.2]. So a E Y + eiR. From Lemma 2, i t fo l lows t h a t
Y + C e.R i s a semiprime i d e a l .
COROLLARY 5. L e t r e . I i E I] b e a s e t of idempotent
e lements o f R.
( i ) N + e.R i s a comple te ly semiprime i d e a l . I n p a r t i c u l a r , g
i f R is a r e g u l a r r i n g and X is a one s i d e d i d e a l then
X is a completely semiprime i d e a l i f and o n l y i f N c X. g -
( i i ) I f N c 2 e .R and e i s r i g h t s e m i c e n t r a l f o r a l l i E I, 3 - 1 i
t h e n e R i s a comple te ly semiprime i d e a l . i
( i i i ) I f Y is a semiprime i d e a l , I i s f i n i t e , and ( 1 - ei)Rei
is a one-sided i d e a l (e.g. i f each e i s r i g h t semi- i
c e n t r a l ) f o r a l l i E I, then Y + C e.R is a semiprime
i d e a l .
COROLLARY 6 . L e t R be a r i n g such t h a t each nonzero reduced
r i g h t i d e a l c o n t a i n s a nonzero idempotent e lement and X a r i g h t
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i d e a l such t h a t N c X. Then N i s DN and t h e r e e x i s t s a s e t of g - g
or thogona l idempotents {ei 1 i E I] such t h a t @ e .R i s reduced
and X is a r i g h t e s s e n t i a l e x t e n s i o n of t h e completely semiprime
i d e a l (G e . R) @ N . 1 g
Proof . It i s a consequence o f [3, P r o p o s i t i o n 1.21 t h a t N g
is DN. L e t {ei I i E I ] b e a s e t of idempotents such t h a t eiR
i s maximal among reduced d i r e c t sums of idempotent genera ted r i g h t
i d e a l s con ta ined i n X [3,p.714] . By [2, Lemma 1 . 3 1 , {ei 1 i E I ]
i s a s e t of o r thogona l idempotents . It f o l l o w s t h a t X is a r i g h t
e s s e n t i a l e x t e n s i o n of (CE eiR) N . T h i s completes t h e p roof . g
Examples of r i n g s which s a t i s f y t h e h y p o t h e s i s of C o r o l l a r y 6
a r e I - r i n g s [g, p.2101, r i n g s w i t h e s s e n t i a l s o e l e , r i g h t
con t inuous r i n g s I?, Theorem 3.81, o r r i n g s f o r which every
s imple r i g h t R-module i s f l a t [g, Lemma 4 1 .
PROPOSITION 7. L e t l e i I i E I ] be a s e t of idempotent
e lements of R such t h a t S = (C eiR) + [n(l - e . ) ~ ] i s an i d e a l .
Then R i s a l e f t e s s e n t i a l e x t e n s i o n of S and ann (S ) _c n r
( I - ei)Ro Furthermore i f each e i s r i g h t s e m i c e n t r a l t h e n R i
is a l s o a r i g h t e s s e n t i a l e x t e n s i o n of S and ann (S) 5 n ( I - e )R . R i
Proof. L e t 0 # x E R . I f S x # O , t h e n R x nS # O . So
assume Sx = 0 . Then e . x = 0 f o r a l l i E I. Hence x E n (1 - ei)R.
Thus R is a l e f t e s s e n t i a l e x t e n s i o n of S and annr(S) 5 0 (1 - e i ) ~ .
For t h e r i g h t s e m i c e n t r a l c a s e t h e above proof i s l e f t - r i g h t
symmetric.
C3ROLLARY 8. L e t rei ( i E 13 b e a s e t r i g h t semicen t ra l
o r thogona l idempotents . Then R i s a l e f t and r i g h t e s s e n t i a l
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COMPLETELY SEMIPRIME IDEALS
ex t ens i on of S = (r eiR) + ["( l - ei)R] and [annr(S)12 =
Proof. Follows from P ropos i t i on 7 and Lemma 3 .
I n t he remainder of t h i s paper we w i l l concen t r a t e on DN
r i g h t i d e a l s and on idempotents i n reduced r i g h t i d e a l s . The
nex t p ropos i t i on shows t h a t every r i g h t i d e a l is e i t h e r reduced
o r con t a in s a nonzero DN r i g h t i d e a l .
PROPOSITION 9. Le t 0 # x E R and n a p o s i t i v e i n t e g e r such
t h a t xn = 0. Then:
( i ) xR con t a in s no nonzero r i g h t s emicen t r a l idempotents.
( i i ) xR i s a DN r i g h t i d e a l .
Proof. ( i ) . Suppose e = x r E xR I s a r i g h t s emicen t r a l
2 idempotent. Now e = e = e x r = exe r = e x ( x r ) r = ex2r2 = .. . =
n n e x r = O .
( i i ) . Le t 0 # x r E xR. I f x r E N, we a r e f i n i s h e d . Assume
m x r 4 N. Thus x rx # 0. By cons ide r i ng x rx , i t can b e shown t h a t
m t h e r e e x i s t s m such t h a t 1 5 rn < n and x rx # 0 , bu t (xrxm) = 0.
Hence xR is DN.
PROPOSITION 10. ( i ) Le t e be a r i g h t s emicen t r a l idempotent
of R such t h a t ( 1 - e)R is a DN r i g h t i d e a l . Then eR
con t a in s an isomorphic copy of every reduced r i g h t i d e a l .
( i i ) L e t (e. I i f 13 be a s e t of r i g h t s emicen t r a l idempotents
of R. I f e .R con t a in s an isomorphic copy of eve ry
reduced r i g h t i d e a l then n ( 1 - e )R i s DN. i
Proof. ( i ) . Le t Y be a reduced r i g h t i d e a l and d e f i n e
f : Y -t eR by f (y) = ey f o r a l l y E P. Clea r ly f i s a R-homomorphism.
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576 BIRKENMEIER
Suppose f (x) = 0 = ex f o r x E Y. Then (,el2 = 0 . Hence xe = 0
because Y is reduced. Thus x = xe + x ( l - e ) = x ( l - e) E ( 1 - e ) ~ .
But Y n (1 - e)R = 0. Therefore ker f = 0. Consequently, f i s
a monomorphism,
( i i ) . Suppose n (1 - e ) R i s not DN. Then t h e r e e x i s t s i
0 # y E n (I - ei)R such t h a t yR i s reduced. Also, t he re e x i s t s
a monomorphism f : yR + Z eiR. Now t h e r e e x i s t s a f i n i t e set
J 5 1 such t h a t f (y) E e R f o r j E J. From Lemma 3 , t he re e x i s t s 3
e = e2 6 2 e,R such t h a t f ( y ) = f (y )e = f ( y e ) . Then y = ye. J
However from Lema 3, ey = 0. Consequently, y2 = (ye12 = 0.
Then y = 0 because yR i s reduced. Contradict ion! Therefore
n (1 - ei)R is DN. This completes the proof.
I n [?I, [i], and [I] the minimal d i r e c t summand conta in ing
the n i l p o t e n t s , denoted MDSN, h a s been s tud ied . The MDSN is a
completely semiprime i d e a l [A, Propos i t ion 1.21.
COROLLARY 11. Let R = A G B where B i s t h e MDSN. Then B is
DN i f and only i f A conta ins an isomorphic copy of every reduced
r i g h t i dea l .
THEOREM 1 2 . Let T be t h e i n t e r s e c t i o n of a l l idempotent
generated r i g h t i d e a l s of R conta in ing N2. Then T i s a completely
semiprime i d e a l which con ta ins every DN r i g h t i d e a l . Moreover, i f
L a = a and aR n T # 0 then aR n N2 # 0,
Proof. Let X be a nonzero DN r i g h t i d e a l , 0 f x E X , and
2 b = b where N c bR. Let e = 1 - b. Then x = ex + bx. We claim
2 - xex = 0. Suppose t h a t xex 0. There e x i s t s r E R such t h a t
xexr $ 0 but (xexr)' = 0 because X i s DN. Therefore ( e ~ r x ) ~ =
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COMPLETELY SEMIPRIME IDEALS 577
e x r ( x e x r ) 2 x = 0. Consequently, exrx = 0 s i n c e eR is reduced.
Hence ( e x r ) 2 = e x r e x r = ( e x r x ) r = 0 because e i s r i g h t s e m i c e n t r a l .
Thus exr = 0 . But t h i s c o n t r a d i c t s t h e f a c t t h a t x e x r # 0.
2 T h e r e f o r e xex = 0 . Consequently (ex) = 0. Hence ex = 0. So
x = bx E bR. It f o l l o w s t h a t X 5 T . By [A, P r o p o s i t i o n 1.21 T is a
completely semiprime i d e a l . L e t 0 # t E aR n T. Then t E ( ~ R ) T .
I f aR i s reduced t h e n T ( 1 - a)R. Hence (aR)T = 0. C o n t r a d i c t i o n !
Thus aR (7 N2 f 0. This completes t h e p roof .
From [A, Theorem 1.41 and Theorem 1 2 , i t can be seen t h a t i f a
r i n g h a s a MDSN t h e n t h e MDSN c o n t a i n s a l l DN r i g h t i d e a l s . When
t h e MDSN is DN t h e n we have an e s p e c i a l l y good decomposi t ion of R
which s e p a r a t e s t h e reduced r i g h t i d e a l s from t h e DN r i g h t i d e a l s
i n t h e s e n s e t h a t t h e MDSN c o n t a i n s a l l DN r i g h t i d e a l s and i t s
complimentary d i r e c t summand of R c o n t a i n s a copy of every reduced
r i g h t i d e a l and i s maximal among reduced r i g h t i d e a l s [A, P r o p o s i t i o n 1.7:
Although no t every r i n g h a s a MDSN, t h e next theorem ensures t h a t
every r i n g has an "approximate" MDSN decomposition.
THEOREM 13. L e t {ei 1 i E I] b e a s e t of idempotent e lements
of R such t h a t e . R is reduced f o r a l l i E I. Then:
( i ) n(1 - e . ) R i s a comple te ly semiprime i d e a l which
c o n t a i n s every DN r i g h t i d e a l .
( i i ) R i s a l e f t and r i g h t e s s e n t i a l e x t e n s i o n of t h e
comple te ly semiprime i d e a l S = (1 e.R) + [>(l - e . )R] .
Furthermore t h e r e e x i s t s a s e t of o r t h o g o n a l idempotents { e . ] j E J ] J
of R such t h a t G e.R is maximal among reduced d i r e c t sums of 3
idempotent genera ted r i g h t i d e a l s and 0 = (@ e.R) n [ n ( l - e . ) R l . 3 3
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578 BIRKENMEIER
Proof . P a r t i i s a consequence of [2, P r o p o s i t i o n 1.21 and
Theorem 12 . P a r t ii is a n a p p l i c a t i o n of Theorem 4 and P r o p o s i t i o n 7 .
The remainder of t h e proof f o l l o w s from [A, p. 7141.
We n o t e t h a t whenever n ( 1 - e . ) R i s DN (e .g. when each
nonzero reduced r i g h t i d e a l c o n t a i n s a nonzero idempoten t ) , then
i t e q u a l s t h e i n t e r s e c t i o n of a l l idempotent genera ted r i g h t i d e a l s
c o n t a i n i n g N 2'
As a n example f o r Theorem 1 3 of a r i n g w i t h no MDSN, l e t
{ R ~ li E I] be a n i n f i n i t e s e t of non-reduced r i n g s each c o n t a i n i n g
a t l e a s t one nonzero reduced idempotent genera ted r i g h t i d e a l (e.g.>Li
i s t h e 2 x 2 upper t r i a n g u l a r m a t r i x r i n g over a f i e l d Fi). L e t V
b e t h e r i n g d i r e c t sum of t h e R Then V1, t h e r i n g wi th u n i t y i'
formed by embedding V i n t h e c a r t e s i a n product of V and t h e
i n t e g e r s , w i l l have t h e d e s i r e d p r o p e r t i e s .
The fo l lowing c o r o l l a r y g e n e r a l i z e s Utumi's decomposition
of a r i g h t con t inuous r e g u l a r r i n g [c, p.6041. We n o t e t h a t i n
a r e g u l a r r i n g every reduced r i g h t i d e a l is a n i d e a l .
COROLLARY 14. L e t R b e a r e g u l a r r i n g where le i 1 i E I]
i s t h e s e t of a l l idempotent e lements of R such t h a t eiR i s reduced
f o r each i E I. Then:
( i ) C e.R i s a reduced i d e a l which c o n t a i n s a l l reduced
i d e a l s and R/C eiR is a DN r i n g .
( i i ) n ( 1 - e , )R is a DN completely semiprime i d e a l which
c o n t a i n s a l l DN r i g h t i d e a l s .
( i i i ) S = (C eiR) G [ n (1 - ei)R i s a completely semiprime
i d e a l which is l e f t and r i g h t f a i t h f u l and R is a l e f t
and r i g h t q u o t i e n t r i n g of S.
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COMPLETELY SEMIPRIME IDEALS
P roo f , ( i ) . By [J, P ropos i t i on 1.21, N 5 n (i - ei)R.
S ince each e is c e n t r a l , t h en 2 e . R i s a reduced i d e a l . C l ea r l y i
Z e .R c o n t a i n s a l l reduced i d e a l s . From [z, p. 681, R I Z eiR is a
DN r i n g . P a r t s ii and iii a r e proved t h e same a s i n Theorem 13 .
This completes t h e proof.
I f R is a l s o r i g h t cont inuous t hen fi ( 1 - e . )R i s t he 1
MDSN [3 , - Theorems 1.4 and 3.91. Thus we have Utumi's decomposition.
F i n a l l y we remark t h a t Co ro l l a ry 14 iii remains t r u e i f n (1 - ei)R
i s rep laced by N . g
ACKNOWLEDGMENTS
This work was supported p a r t l y by NSERC g r a n t #A4033
a t McMaster Un ive r s i t y and p a r t l y by a g r an t from Southeas t
Missour i S t a t e Univers i ty .
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