Modules which are invariant under monomorphisms of their injective hulls


Alahmadi A., Er N. , Jain S.

JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, vol.79, pp.349-360, 2005 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 79
  • Publication Date: 2005
  • Doi Number: 10.1017/s1446788700010946
  • Title of Journal : JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY
  • Page Numbers: pp.349-360

Abstract

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of R into R-R((N)) can be extended to R-R. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is quasi-injective. The following results are obtained: M is quasi-injective if and only if M is pseudo-injective and M-2 is CS. Furthermore, if M is a direct sum of uniform modules, then M is quasi-injective if and only if M is pseudo-injective. As a consequence of this it is shown that over a right Noetherian ring R, quasi-injective modules are precisely pseudo-injective CS modules.