An approach to extending modules via homomorphisms

Abstract

The notion of $mathcal{K}$-extending modules was defined recently as a proper generalization of both extending modules and Rickart modules. Let $M$ be a right $R$-module and let $S=End_R(M)$. We recall that $M$ is a $mathcal{K}$-extending module if for every element $phiin S$, $Kerphi$ is essential in a direct summand of $M$. Since a direct sum of $mathcal{K}$-extending modules is not a $mathcal{K}$-extending module in general, an open question is to find necessary and sufficient conditions for such a direct sum to be $mathcal{K}$-extending. In this paper, we give an answer to this question. We show that if $M_i$ is $M_j$-injective for all $i, jin I ={1, 2, dots, n}$, then $bigoplus_{i=1}^n M_i$ is a $mathcal{K}$-extending module if and only if $M_i$ is $M_j$-$mathcal{K}$-extending for all $i, j in I$. Other results on $mathcal{K}$-extending modules and some of their applications are also included.