On semilocal rings

Zhang, Hongbo

doi:10.1090/S0002-9939-08-09577-4

On semilocal rings
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by Hongbo Zhang

Proc. Amer. Math. Soc. 137 (2009), 845-852

DOI: https://doi.org/10.1090/S0002-9939-08-09577-4

Published electronically: September 17, 2008

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Abstract:

In this paper, semilocal rings are characterized in different ways; in particular, it is proved that a ring $R$ is semilocal if and only if every descending chain of principal right ideals of $R$, $a_0R\supseteq a_1R\supseteq a_2R\supseteq \cdots \supseteq a_nR\supseteq \cdots \text { with }a_{i+1}=a_i-a_ib_ia_i$ eventually terminates. Then modules with semilocal endomorphism rings are characterized by chain conditions.

References

Rosa Camps and Warren Dicks, On semilocal rings, Israel J. Math. 81 (1993), no. 1-2, 203–211. MR 1231187, DOI 10.1007/BF02761306
Alberto Facchini, Module theory, Progress in Mathematics, vol. 167, Birkhäuser Verlag, Basel, 1998. Endomorphism rings and direct sum decompositions in some classes of modules. MR 1634015
Alberto Facchini, Dolors Herbera, Lawrence S. Levy, and Peter Vámos, Krull-Schmidt fails for Artinian modules, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3587–3592. MR 1277109, DOI 10.1090/S0002-9939-1995-1277109-4
Dolors Herbera and Ahmad Shamsuddin, Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3593–3600. MR 1277114, DOI 10.1090/S0002-9939-1995-1277114-8
C. Lomp, On dual Goldie dimension, Diplomarbeit, Heinrich Heine Universität, Düsseldorf, 1996.
K. Varadarajan, Dual Goldie dimension, Comm. Algebra 7 (1979), no. 6, 565–610. MR 524269, DOI 10.1080/00927877908822364