A note on modules (Q1123246)

A submodule H of M is said to be E-irreducible if whenever \(H=K\cap J\) where K and J are submodules of M and H is essential in K, then either \(H=K\) or \(H=J\). Every complement submodule is an E-irreducible submodule, but the converse is not true. The purpose of this note is to characterize complement submodules by means of E-irreducibility. It is shown that for a submodule K of an R-module M, K is a complement if and only if either \(K=M\) or K is not essential, but is E-irreducible.