Rad-supplementing modules
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초록
Let $R$ be a ring, and let $M$ be a left $R$-module. If $M$ is Rad-supplementing, then every direct summand of $M$ is Rad-supplementing, but not each factor module of $M$. Any finite direct sum of Rad-supple\-menting modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. $M$ has a Rad-supplement in its injective envelope if and only if $M$ has a Rad-supplement in every essential extension. $R$ is left perfect if and only if $R$ is semilocal, reduced and the free left $R$-module $(_R R)^{(\mathbb{N})}$ is Rad-supplementing if and only if $R$ is reduced and the free left $R$-module $(_R R)^{(\mathbb{N})}$ is ample Rad-supplementing. $M$ is ample Rad-supplementing if and only if every submodule of $M$ is Rad-supplementing. Every left $R$-module is (ample) Rad-supplementing if and only if $R/P(R)$ is left perfect, where $P(R)$ is the sum of all left ideals $I$ of $R$ such that $\Rad I = I$.
Abstract
Let $R$ be a ring, and let $M$ be a left $R$-module. If $M$ is Rad-supplementing, then every direct summand of $M$ is Rad-supplementing, but not each factor module of $M$. Any finite direct sum of Rad-supple\-menting modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. $M$ has a Rad-supplement in its injective envelope if and only if $M$ has a Rad-supplement in every essential extension. $R$ is left perfect if and only if $R$ is semilocal, reduced and the free left $R$-module $(_R R)^{(\mathbb{N})}$ is Rad-supplementing if and only if $R$ is reduced and the free left $R$-module $(_R R)^{(\mathbb{N})}$ is ample Rad-supplementing. $M$ is ample Rad-supplementing if and only if every submodule of $M$ is Rad-supplementing. Every left $R$-module is (ample) Rad-supplementing if and only if $R/P(R)$ is left perfect, where $P(R)$ is the sum of all left ideals $I$ of $R$ such that $\Rad I = I$.
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