Modules invariant under monomorphisms of their envelopes


Guil Asensio P. A. , KESKİN TÜTÜNCÜ D. , Srivastava A. K.

ADVANCES IN RINGS AND MODULES, vol.715, pp.171-179, 2018 (Journal Indexed in SCI) identifier

  • Publication Type: Article / Article
  • Volume: 715
  • Publication Date: 2018
  • Doi Number: 10.1090/conm/715/14410
  • Title of Journal : ADVANCES IN RINGS AND MODULES
  • Page Numbers: pp.171-179

Abstract

In this paper we prove that a module M is invariant under automorphisms of its X-envelope if and only if M is invariant under monomorphisms of its X-envelope where X is an enveloping class with the property that for each object X in X, End(X)/J(End(X)) is a von Neumann regular right self-injective ring and idempotents lift modulo J(End(X)). In particular, invariance under monomorphisms is equivalent to the invariance under automorphisms of X-envelope when X is the class of injective modules, pure-injective modules or flat cotorsion modules. We finish the paper by proving that an X-automorphism invariant (equivalently, X-monomorphism invariant) module M satisfies the property P if and only if its X-envelope X(M) satisfies the property P, when P is direct-finiteness, internal cancellation, cancellation or the substitution property.