Modules closed full large extensions of cyclic submodules are summands

Abstract

This paper introduced a new generalization of extending modules, namely modules in which every closed full large extension of a cyclic submodule is a direct summand, introduced a new generalization of the concept of injective modules.  In fact, we give and study the properties of the concept of full-LE-Cy-injective modules.  Although full-LE-Cy-injective modules are far from injective modules, they are exactly the same on some kind of rings.  Then we make use of relatively full-LE-Cy-injectivity on modules to study direct sums of two (C₁-LE-Cy)-modules.  We show that a direct sum of two relatively full-LE-Cy-injective modules is a (C₁-LE-Cy)-module if and only if each one of them is a (C₁-LE-Cy)-module.  Examples are provided to illustrate and delimit the theory.