an approach to extending modules via homomorphisms

The notion of k-extending modules was defined recently as a proper generaliza- tion of both extending modules and rickart modules. let m be a right r-module and let s = endr(m ). we recall that m is a k-extending module if for every element φ ∈ s, ker φ is essential in a direct summand of m . since a direct sum of k-extending modules is not a k-extending module in general, an open question is to find necessary and sufficient conditions for such a direct sum to be k-extending. in this paper, we give an answer to this question. we show that if mi is mj -injective for all i, j ∈ i = {1, 2, . . . , n}, then ⊕n mi is a k-extending module if and only if mi is mj -k-extending for all i, j ∈ i. other results on k-extending modules and some of their applications are also included.