A limit on the Loewy length of the endomorphism ring of a module of finite length
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- by Sverre O. Smalø PDF
- Proc. Amer. Math. Soc. 81 (1981), 164-166 Request permission
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 ${n_i}$, $i = 1, \ldots ,m$.References
- Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0417223, DOI 10.1007/978-1-4684-9913-1
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 81 (1981), 164-166
- MSC: Primary 16A65; Secondary 16A64
- DOI: https://doi.org/10.1090/S0002-9939-1981-0593447-2
- MathSciNet review: 593447