An approach to extending modules via homomorphisms

The notion of -extending modules was defined recently as a proper generalization of both extending modules and Rickart modules. Let be a right -module and let . We recall that is a -extending module if for every element , is essential in a direct summand of . Since a direct sum of -extending modules is not a -extending module in general, an open question is to find necessary and sufficient conditions for such a direct sum to be -extending. In this paper, we give an answer to this question. We show that if is -injective for all , then is a -extending module if and only if is --extending for all . Other results on -extending modules and some of their applications are also included.