Modules which are invariant under monomorphisms of their injective hulls

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Alahmadi A., Er N., Jain S.

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


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.