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.
Published Version
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