An approach to extending modules via homomorphisms

Document Type : Research Paper


Department of Mathematics, Quchan University of Technology, P.O.Box 94771-67335, Quchan, Iran


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 $\phi\in S$, $Ker\phi$ 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 $i, j\in I =\{1, 2, \dots, n\}$, 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. 


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