Abstract
The main purpose of this work is to introduce and study the concept of pseudo-rc-injective module which is a proper generalization of rc-injective and pseudo-injective modules. Numerous properties and characterizations have been obtained. Some known results on pseudo-injective and rc-injective modules generalized to pseudo-rc-injective. Rationally extending modules and semisimple modules have been characterized in terms of pseud-rc-injective modules. We explain the relationships of pseudo-rc-injective with some notions such as Co-Hopfian , directly finite modules.
Highlights
Throughout, R represent an associative ring with identity and all R-modules are unitary right modules.Let M and N be two R-modules, N is called pseudo –M- injective if for every submodule Aof M, any Rmonomorphismf: A → N can be extended to an R-homomorphism α: M → N
In the following result we show that, for a uniform R-module the concepts of the rc-injective modules and pseudorc-injective are equivalents
In the following proposition we show that the co-Hopfian and directly finite R-modules are equivalent under pseudo-rc-injective property
Summary
Let M and N be two R-modules, N is called pseudo –M- injective if for every submodule Aof M, any Rmonomorphismf: A → N can be extended to an R-homomorphism α: M → N. Definition 2.1Let M and N be two R-modules . N is pseudo M-rationally closed-injective ( briefly pseudo M-rcinjective ) if for every rationally closed submodule H of M , any R-monomorphism φ: H → N can be extended to an R– homomorphism β: M → N . (⟸) Suppose that M is a pseudo-rc- injective, let K be rationally closed submodule of M and α ∶ K → M be R-
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