Generalizations of injective modules: Red-Injective and strongly Red-Injective modules.

Abstract

As generalizations of injective modules, Red-injective and strongly Red-injective modules are introduced. The whole study is based on extensive use of de nitions, propositions and theorems. Properties of semi-Artinian, quasi-Frobenius and right V -rings have provided a basis for other properties so derived. Many properties of Red-injective and strongly Red-injective modules are derived. Among them, there are: (1) The class of Red-injective modules is closed under direct products and summands. (2) A semi-simple module is Soc-injective if and only if it is Red-injective. (3) Over a Principal Ideal Domain (P.I.D), every projective module is Red-injective if and only if every free module is Red-injective. (4) For a Noetherian module MR, any direct sum of Red-M-injective modules is Red-injective. (5) Quasi-Frobenius and right V -rings are characterised in terms of strongly Red-injective modules. It is shown that an injective module is strongly Red-injective, a strongly Red-injective module is strongly Soc-injective, and a strongly Soc-injective module is strongly min-injective. Furthermore, it is shown that Red-injectivity is not a Morita invariant property.