On semilocal rings

Zhang, Hongbo

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

On semilocal rings
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Proc. Amer. Math. Soc. 137 (2009), 845-852 Request permission

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.

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