J. Aust. Math. Soc.  79 (2005), 349-360
Modules which are invariant under monomorphisms of their injective hulls

A. Alahmadi
  Department of Mathematics
  Ohio University
  Athens, OH 45701
  USA
  noyaner@yahoo.com
N. Er
  Department of Mathematics
  The Ohio State University-Newark
  OH 43055
  USA
 
and
S. K. Jain
  Department of Mathematics
  The Ohio State University-Newark
  OH 43055
  USA
 


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 $R_R^{(\mathbb{N})}$ 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.
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