Advances in Non-Commutative Ring Theory: Proceedings of the by Patrick J. Fleury (auth.)

By Patrick J. Fleury (auth.)

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Extra resources for Advances in Non-Commutative Ring Theory: Proceedings of the Twelfth George H. Hudson Symposium Held at Plattsburgh, USA, April 23–25, 1981

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Is left stable, HomR(E(N),H ) = 0. HOmRp((E(N)p,H) = 0. envelope of N. 61], By [6, Prop. I], (E(N))p is the injective Therefore Rp is a left stable ring. It is well known that R is FLBN if and only if there exists a I-i correspondence between the prime ideals and the indecomposable injective R-modules [29, Thm. VII. 1]. Since left localization at P is a perfect localization, a i-i correspondence exists between the prime ideals of Rp and the indecomposable injective Rp-modules [i0, Prop. 14]. Therefore Rp is FLBN.

6 every 1 of K a t o applies. 8 i__ssr i g h t Corollary. FGF, then If Q Q is a c o g e n e r a t o r is QF. Thus, any left o f Q-mod, PF r i g h t and if Q FGF r i n g i_ss Any PF Q_£F. ring Proof. The is left cogenerating, 4. RIGHT Right by J a i n injective Colby first FGF IF r i n g s [73]. part RINGS have (Briefly, right R-module follows from so the s e c o n d ARE been the part theorem. left follows. IF studied a ring R is flat. ) For a proof of the next result see L e m m a 20 of R u t t e r [74].

Natural is r i g h t TF, states Theorem. IN P R O J E C T I V E S . 1 in -- of of m o d - R , is t o r ~ i o n l e s s , if this. R MODULES is, Q = Q(R) for every a cogenerator of 477, the e x p o n e n t case w h e n first torsionless that asking when The the m o r e [60, p. left Noetherian, (not TORSIONLESS ( ,R) an e m b e d d i n g M --~ R n M** --+ Rn). A module M I is an e m b e d d i n g . induces R Clearly, is t o r s i o n l e s s iff the canonical map -~ R M ( if ,f(m) .... ,f , 1 n then left of h converts R Rn--~ exact exactness ---~ 0 M ~ into 0 --~ M** --~ R n exact.

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