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Characterizing $S$-projective modules and $S$-semisimple rings by uniformity

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 Added by Xiaolei Zhang
 Publication date 2021
  fields
and research's language is English




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Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called $S$-projective provided that the induced sequence $0rightarrow {rm Hom}_R(P,A)rightarrow {rm Hom}_R(P,B)rightarrow {rm Hom}_R(P,C)rightarrow 0$ is $S$-exact for any $S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $S$-projective modules are obtained. The notion of $S$-semisimple modules is also introduced. A ring $R$ is called an $S$-semisimple ring provided that every free $R$-module is $S$-semisimple. Several characterizations of $S$-semisimple rings are provided by using $S$-semisimple modules, $S$-projective modules, $S$-injective modules and $S$-split $S$-exact sequences.



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83 - Xiaolei Zhang 2021
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $T$ is called uniformly $S$-torsion provided that $sT=0$ for some $sin S$. The notion of $S$-exact sequences is also introduced from the viewpoint of uniformity. An $R$-module $F$ is called $S$-flat provided that the induced sequence $0rightarrow Aotimes_RFrightarrow Botimes_RFrightarrow Cotimes_RFrightarrow 0$ is $S$-exact for any $S$-exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. A ring $R$ is called $S$-von Neumann regular provided there exists an element $sin S$ satisfies that for any $ain R$ there exists $rin R$ such that $sa=ra^2$. We obtain that a ring $R$ an $S$-von Neumann regular ring if and only if any $R$-module is $S$-flat. Several properties of $S$-flat modules and $S$-von Neumann regular rings are obtained.
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105 - Xiaolei Zhang 2021
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