On the Endomorphism Ring of a Noetherian Chain Module

The aim of this paper is to further develop the theory of the well-known Schur's lemma on endomorphism rings for application to wider structures. We show that for a commutative Noetherian chain ring R and Noetherian chain modules M and N, the left R-module HomR (M,N) will be a chain module. Then, we conclude that the endomorphism ring EndR (M) of the Noetherian chain module M over such R is a Noetherian chain ring. As a special case, we consider that the dual R- module M * = HomR (M,R) of the Noetherian chain R-module M is a Noetherian chain module.