Schmidt Modules and Some of Their Applications

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.