Endoproperties of modules and local duality

JOURNAL OF ALGEBRA

卷:316

Article

Dung, Nguyen Viet ; Garcia, Jose Luis

关键词: local duality;    endonoetherian module;    endofinite module;    pure semisimple ring;    ring of finite representation type;   

DOI  :  10.1016/j.jalgebra.2007.06.029

来源: Elsevier

【 摘 要 】

Let R be any ring and N = circle plus(i is an element of I) N-i be a direct sum of finitely presented left R-modules N-i. Suppose that D(N) and D(N-i) are the local duals of N and Ni for each i is an element of I. We prove that the lattice of endosubmodules of N is anti-isomorphic to the lattices of matrix subgroups of D(N) and of M = circle plus(i is an element of I) D(N-i). As consequences, N is endoartinian if and only if M (or D(N)) is endonoetherian, and N is endonoetherian if and only if M (or D(N)) is Sigma-pure-injective. We obtain, in particular, that if R is a Krull-Schmidt ring, and M is an indecomposable, pure-injective endonoetherian right R-module which is the source of a left almost split morphism in Mod(R), then M is endofinite. As an application, a ring R is of finite representation type if and only if every pure-injective right R-module is endonoetherian. (C) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free