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Download e-book for kindle: Graded Ring Theory by Constantin Năstăsescu, F. van Ostaeyen

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By Constantin Năstăsescu, F. van Ostaeyen

ISBN-10: 044486489X

ISBN-13: 9780444864895

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Example text

E . au i s i n n e r and i n d u c e d by ( c ( ~ ) ) - ' . 2. 3. 3. i t f o l l o w s 2 i n d e e d d e t e r m i n e a n eleinent o f H (H, U ( ( Z ( R ) ) e ) ) . I f p r o d u c t theorem. Because o f t h e remarks t h a t do,Tdoes t h e Yo a r e r e p l a c e d by u ( ~ ) Yf o~ r c e r t a i n u ( ~ )C U((Z(R)),) the c ( ~ ) 41 1. Graded Rings and Modules dU,T w i l l be r e p l a c e d by an e q u i v a l e n t 2 . 7. 4. L e t R be a s t r o n a l y graded d i v i s i o n r i n g o f t y p e Proposition.

I f P i s a prime i d e a l o f Re then Po i s a prime i d e a l of Re f o r a l l u C G . I f I and J a r e i d e a l s of Re such t h a t I J c P' -1 = 1o-l Ja - l (P')5-1 = P , hence I 0 c P say. Proof. Then I c P' By then ( I J ) ' -1 = follows. Prim ( - ) we denote t h e s u b s e t of Spec ( - ) c o n s i s t i n g of the p r i m i t i v e i d e a l s of t h e r i n g under c o n s i d e r a t i o n . 32. C o r o l l a r y . Jacobson r a d i c a l of Re, J ( R e ) , and the n i l r a d i c a l o f Re, rad(Re), a r e both G - i n v a r i a n t .

Conversely, any element o f J ( A where p ( T ) F k [ T I + 69 k That I @ k(T) c J(A k ( T ) ) i s obvious k k ( T ) ) can be w r i t t e n as D ( T ) - I (amTm+.. @ i s nonzero and ai F A. 14 . k we o b t a i n f o r a l l n > m : n ai T’ F J ( A 6n k ( T ) ) ; t h u s ai T’ f J ( A qh k ( k k and t h e r e f o r e ai F I . Pick x c I, t h e n ( 1 + xT)-’ t h e c o e f f i c i e n t s have t o = 1 - xT + x2T2 -... i n A 60 k ( ( T ) ) . However k l y i n a f i n i t e d i m e n s i o n a l k-subspace o f A i .

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Graded Ring Theory by Constantin Năstăsescu, F. van Ostaeyen


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