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Page 1: Idempotents and completely semiprime ideals

This article was downloaded by: [Moskow State Univ Bibliote]On: 06 November 2013, At: 20:51Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

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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

To link to this article: http://dx.doi.org/10.1080/00927878308822865

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Page 2: Idempotents and completely semiprime ideals

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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574 BIRKENMEIER

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 .

REFERENCES

1. V. A. Andrunakievic and Ju.M. Rjabuhn, Rings without n i l p o t e n t elements , and completely s imple i d e a l s , Sovie t Mat. Dokl. 2 ( l968) , 565-568.

2. G. F. Birkenmeier, A decomposition theory of r i n g s , Ph.D. Thes i s , Un ive r s i t y of Wisconsin-Milwaukee, Milwaukee, Wisconsin, 1975.

3. , S e l f - i n j e c t i v e r i n g s and t h e minimal d i r e c t summand con t a in ing t h e n i l p o t e n t s , Comm. i n Algebra 4(1976), 705-721. -

4. , Indecomposable decompositions and t he minimal d i r e c t summand con t a in ing t he n i l p o t e n t s , Proc. h e r . Math. Soc. E ( l 9 7 9 ) , 11-14,

5. , Baer r i n g s and quasi-continuous r i n g s have a MDSN, P a c i f i c J. Math., t o appear .

6 . N. J. Divinsky, Rings and Radica ls , Un ive r s i t y of Toronto P re s s , Toronto, Ont., 1965.

Dow

nloa

ded

by [

Mos

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Sta

te U

niv

Bib

liote

] at

20:

51 0

6 N

ovem

ber

2013

Page 15: Idempotents and completely semiprime ideals

5 80 BIRKENMEIER

7. C. F a i t h , I n j e c t i v e quo t i en t r i n g s of commutative r i n g s , i n Module Theory, Spr inger Lec tu r e Notes No. 700, Spr inger Verlag, B e r l i n , 1979.

8 . J. W. F i s h e r , On t h e n i l po t ency of n i l subr ings , Can. J. Math. Z ( l 9 7 O ) , 1211-1216.

9. K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Pure and Applied Mathematics S e r i e s , Vol. 33, Marcel Dekker, New York, 1976.

10. N. Jacobson, S t r u c t u r e of Rings, Colloquium Pub l i c a t i on Vol. 37 Amer. Math, Soc., Providence 1955, 1964.

11. L. Jeremy, Modules e t anneaux quasi-continus, Canad. Math. Bul l . g ( 2 ) ( l 974 ) , 217-228.

12. J . Lambek, Lec tu r e s on Rings and Modules, B l a i s d e l l , Waltham, Mass., 1966.

13. N. H. McCoy, Completely prime and completely semi-prime i d e a l s , i n Rings, Modules and Radica ls , Col loquia Mathematica S o c i e t a l i s Janos Bolya i , Vol. 6 , North-Holland, Amsterdam, 1973.

14 . R. C. Shock, E s s e n t i a l l y n i l p o t e n t r i n g s , I s r a e l J. Math - 9 (1971), 180-185.

15. G. T h i e r r i n , Sur l e s ideaux completement premiers d'un anneau quelconque, Bul l . Acad. Roy. Belg., G ( l 9 5 7 ) , 124-132.

16. , On duo r i n g s , Can. Math. Bul l . , 3(1960), 167-172. -

17. Y. Utumi, On continuous r e g u l a r r i n g s and semi-simple s e l f - i n j e c t i v e r i n g s , Canad. J. Math g ( 1 9 6 0 ) , 597-605.

18. E. T. Wong, Regular r i n g s and i n w g r a l ex t ens ion of a r e g u l a r r i n g , Proc. Amer. Math. So?. 2 ( 1 9 7 2 ) , 313-315.

19. R. Yue Chi Ming, On r e g u l a r r i n g s and Ar t i n i an r i n g s , Riv. Mat. Univ, Parma, t o appear.

Received: November 1981

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