A limit on the Loewy length of the endomorphism ring of a module of finite length

Abstract:

Let $M$ be a module of finite length over an arbitrary ring $A$. Let ${S_1},{S_2}, \ldots ,{S_m}$ be the nonisomorphic simple composition factors of $M$ and let $[M] = {n_1}[{S_1}] + {n_2}[{S_2}] + \cdots + {n_m}[{S_m}]$ denote that ${S_i}$ occurs ${n_i}$ times in a composition series for $M$. As a generalization of Schur’s lemma we have the following wellknown result: If $M$ is nonzero and all the ${n_i}$ are equal to one then the Loewy length of the endomorphism ring of $M$ is one. We will generalize this fact showing that in general the Loewy length of the endomorphism ring of $M$ is less than or equal to the maximum of the ${n_i}$, $i = 1, \ldots ,m$.