Direct Sums of Extending Modules

Abstract

Let M and N be left R-modules. N is called M -ep-injective if N is essentially M -injective and pseudoly M -injective. An example is given to show that M -ep-injectivityis a non-trivial generalization of M -injectivity. We show that any direct sum of relatively ep-injective extending left R-modules of finite length is special extending. It is also proved that the direct sum M = i2 I M i of left R-modules M i (i 2 I,jIj 2) is extending if and only if there exist i 6= j in I such that every closed submodule K of M with K \ M i e Kor K \ M j e K or K \ M i = K \ M j = 0 is a direct summand.