By Patrick J. Fleury (auth.)
Read or Download Advances in Non-Commutative Ring Theory: Proceedings of the Twelfth George H. Hudson Symposium Held at Plattsburgh, USA, April 23–25, 1981 PDF
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Extra info for Advances in Non-Commutative Ring Theory: Proceedings of the Twelfth George H. Hudson Symposium Held at Plattsburgh, USA, April 23–25, 1981
PROOFS OF THEOREMS. 1 Theorem. q u o t i e n t rin@ Q. Let R Then, be a left and right Ore tin@ with R is left and ri@ht TF iff Q is Q_~. 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.
5c Q is right Noetherian. 6 Q is right selfinjective. (3,7-8 is QF. 7) corresponding Q the implication R right TF ===~ Q or equivalently, is TF iff " if Q is commutative) on annihilator right . . . . Every left ideal). , ideal left ideals. ideals. right Noetherian. right socle. 6) author's knowledge 2. right is a result NOTATION in other embeds Q = Q(R) the problems AND BACKGROUND. 1) that Levy's FGF = ~ right that a ring Since Q To the are open. generated) A is right every module is right TF q u e s t io n A .
We first claim that Ext~(R/P,R/J) Ext~(R/P,R/J) (1) 0. Suppose Since R is stable with unique simple submodule injective resolution Ej = • E(S) for all j. of R/J has the property that Let 0 ~ ci E Ext~(R/P,R/J) = ker d~/Im d~_ 1 • Pick ~ E HomR(R/P,E k) such that ~p E ker d~ and ~ + Im d~_ 1 = ~. Since E k = @ E(S) and J has the Artin-Rees finite length power of x [13, Prop. 3]. such that x ~ ( R / P ) property, In particular, = 0. Consider R~(R/P) there exists a the exact sequence ~+i 0 ~ R/P x__+ R/P ~ R/Rx~+I+P ~ 0 The functor Ext (-,R/J) ....