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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Sverre O. Smalø
Journal: Proc. Amer. Math. Soc. 81 (1981), 164-166
MSC: Primary 16A65; Secondary 16A64
MathSciNet review: 593447
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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$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0593447-2
PII: S 0002-9939(1981)0593447-2
Article copyright: © Copyright 1981 American Mathematical Society



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