On the Endomorphism Ring of a Noetherian Chain Module
Hanan A. Alolaiyan *
Department of Mathematics, Faculty of Sciences, King Saud University, Riyadh, Saudi Arabia.
Ahmad M. Alghamdi
Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Alqura University, P.O.Box 14035, Makkah 21955, Saudi Arabia.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Chain rings, chain module, Noetherian ring, Noetherian module, endomorphism ring, Schur's lemma.