An approach to extending modules via homomorphisms

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 =\{۱, ۲, \dots, n\}, then \bigoplus_{i=۱}^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. 

کلیدواژه ها:

Extending module ، Endomorphism ring ، Rickart module ، Semiregular ring