Projectively coresolved Gorenstein flat modules over trivial ring extensions

Le résumé fourni par la source

summary:Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$. Sufficient and necessary conditions are established for projectively coresolved Gorenstein flat (PGF, for short) modules over $R\ltimes M$. More pecisely, it is proved that $(X, \alpha )$ is a PGF left \hbox {$R\ltimes M$-module} if and only if ${\rm Coker}(\alpha )$ is a PGF left $R$-module and the sequence $M\otimes _{R}M\otimes _{R}X \overset {M\otimes \alpha } \to \longrightarrow M\otimes _{R}X \overset {\alpha } \to \longrightarrow X$ is exact under some assumptions on $M$. As applications, it is characterized PGF modules over Morita rings with zero bimodule homomorphisms.

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