The endomorphism ring of an Artinian module whose homogeneous length is finite

Schulz, Rainer

doi:10.1090/S0002-9939-1982-0667274-2

The endomorphism ring of an Artinian module whose homogeneous length is finite
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Proc. Amer. Math. Soc. 86 (1982), 209-210 Request permission

Abstract:

Smalø[2] showed that the index of nilpotency of the endomorphism ring of a module ${M_R}$ of finite length is bounded by the number $\max \left \{ {{n_A}|{A_R}\;{\text {simple}}} \right \}$, where ${n_A}$ denotes the number of times ${A_R}$ occurs as a factor in a composition chain of ${M_R}$. We give another proof of Smalø’s theorem which leads to an analogous result for artinian modules whose homogeneous length is finite.

References

Joe W. Fisher, Nil subrings of endomorphism rings of modules, Proc. Amer. Math. Soc. 34 (1972), 75–78. MR 292878, DOI 10.1090/S0002-9939-1972-0292878-2
Sverre O. Smalø, A limit on the Loewy length of the endomorphism ring of a module of finite length, Proc. Amer. Math. Soc. 81 (1981), no. 2, 164–166. MR 593447, DOI 10.1090/S0002-9939-1981-0593447-2