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Mat. Zametki, 2008, Volume 84, Issue 5, Pages 681–692 (Mi mz4164)  

Schmidt Modules and Some of Their Applications

V. A. Vedernikova, N. V. Yakubovskijb

a Moscow City Pedagogical University
b Moscow State Pedagogical University

Abstract: Let $R$ be an associative ring with unit. A nonsemisimple right $R$-module $M=M_R$ is referred to as a (right) Schmidt module if every proper (right) submodule in $M$ is semisimple, and a module $M$ is called a (right) generalized Schmidt module if $M$ is not a Schmidt module and each of its proper (right) submodule is either a semisimple module or a Schmidt module. A left Schmidt $R$-module and a left generalized Schmidt $R$-module are defined similarly. In the paper, a complete description of the structure of right Schmidt $R$-modules and generalized Schmidt $R$-modules is given, the existence of Schmidt $R$-submodules in any nonsemisimple Artinian module is established, and a complete description of nonsemisimple Artinian modules in which every Schmidt submodule is distinguished as a direct summand is presented. As corollaries, characterizations of (generalized) Schmidt modules over a Dedekind ring and over a matrix ring over this ring are obtained in the paper.

Keywords: Schmidt module, generalized Schmidt module, associative ring, Dedekind ring, semisimple module, Artinian module, local module, Morita equivalence

DOI: https://doi.org/10.4213/mzm4164

Full text: PDF file (468 kB)
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English version:
Mathematical Notes, 2008, 84:5, 636–645

Bibliographic databases:

UDC: 512.553.1
Received: 27.12.2005
Revised: 04.04.2008

Citation: V. A. Vedernikov, N. V. Yakubovskij, “Schmidt Modules and Some of Their Applications”, Mat. Zametki, 84:5 (2008), 681–692; Math. Notes, 84:5 (2008), 636–645

Citation in format AMSBIB
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