By John Riordan
ISBN-10: 0471722758
ISBN-13: 9780471722755
ISBN-10: 0882758292
ISBN-13: 9780882758299
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Extra info for Combinatorial Identities
Example text
C. 2], ring. 7]. 9) completes the proof. ; Necessity of the condition is clear. i ii: ! Section 6; i In this section we collect a pair of pretty results, one due to Posner, the other to Procesi. 1). I. algebra over i t s f i e l d is a sum of nilpotent elements, F . If A must be n i l . i ' tI Proof: We show that A has no prime ideals. Suppose i s a prime ideal of A P nilpotent ideals and hence P . Then i s an algebra ideal. I. algebra with the property that every element of A/P is a sum of nilpotent elements.
C. C. on r i g h t a n n i h i l a t o r s so t h a t t h e r e i s an i n t e g e r B = {x e A : xA If Now fiB < B for x j 0 n = 0} B = A n and l e t , A such t h a t A = A/B . Also, A A n = 0 (x e A) . Define ' j . i s nilpotent. and hence we see t h a t xA Therefore, suppose i s a n i l r i n g i n which s a t i s f i e s the same i d e n t i t i e s as A . B j= A xA j= 0 Thus, , 1 53. if . t n i l p o t e n t we may reduce t o t h e case i n w h i c h x = 0 xA = 0 only . \ To show t h a t t h i s i s i m p o s s i b l e we assume t h a t A satisfies } j the m u l t i l i n e a r i d e n t i t y Let From Ae = 0 V i f ( x a J E( \ a 2 E X a x a a(2) CF(k) s ••• > i s of degree i .
E . , where J , and J A/U i s right M primitive. i s a maximal modular r i g h t i d e a l , i s a maximal modular l e f t i d e a l . u " = { a e A: aM' =0} k F o r , i f aM" = 0 k or Define • >'< M' = A / J Claim: A-module w h i c h i s ; : * - f a i t h f - u l . a e U k aA <_ J . Now l e t , aM' = 0 a e U , or k aA <_ J , then k Aa k . e. k a e U , Ma = 0,Aa ^ _ J , i . e . a e { a e A : aM" = 0} M' i s a faithful irreducible left F i n a l l y , i f a e U f\ U* , then k From aM' = 0 e U k 1 k Hence a we have aA _< J Ma = 0 k , (A/U )-module.
Combinatorial Identities by John Riordan
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