Download e-book for kindle: Classic Algebra by P. M. Cohn

By P. M. Cohn

ISBN-10: 047187731X

ISBN-13: 9780471877318

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2 then Q Theorem. is QF. If Q is a riqht A r t i n i a n ri@ht FGF ring, 29 Proof. 5, to prove in order that Q it suffices to show that: cogenerator (Faith-Walker to prove is right Q that Q is QF, selfinjective. 1]). ,n. Then, C = E ~ -'' ~ E is the least 1 1 n injective c o g e n e r a t o r of mod-Q. Let E denote any of the E , 1 and let F be any finitely g e n e r a t e d s u b m o d u l e of E. Then, by the FGF assumption, finite generation smallest nonzero generated hence C = E E E E F can be assumed However, submodule ~ ''' ~ E 1 n cogenerator.

6] If R is a classically local noetherian ring with unique maximal ideal J, then the J-adic completion of R is ring isomorphic to the bicommutator of E(S) where S is the unique simple left R-module. Corollary i0: with unique maximal = If R is a classically local noetherian ring ideal J and unique simple R-module S such that E(S) is artinian, then R is left noetherian. Furthermore, E(S) produces a Morita duality between the rings R and T ~End(E(S)) Proof: duality. By M~ller [22, Thm. 8], E(S) produces a Morita Therefore E(S) T is artinian = HomT(E(S),E(S)) Corollary ii: is left noetherian [i, Thm.

Simple module. dk_ 1 = ker Although Suppose local, of the unique envelope Let T k = HomR(R/J,Ek) • = 0. classically injective of M. R-homomorphism We claim that dk(Tk) Since resolution Let If M is a then ExtkR (M, lim N~)=~ lira Extk(M,Na) Proof: We prove 0 ~K~P be an exact R-module. generated. sequence the result by induction on k. Let ~M~0 with P, a finitely Since R is left noetherian, The diagram generated K is also projective left finitely 53 HOmR(P,l~m N ) ~ HOmR(K,l~m N ) ~ Ext~(M,l~m N ) ~ 0 15 18 Iv l~m HomR(P,N ) ~ l~m HomR(K,N ) ~ l~m Ext~(M,Na) ~ 0 is exact with ~ and p being isomorphisms the result follows for k = i.

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Classic Algebra by P. M. Cohn

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