Frobenius扩张上的$n$-Gorenstein投射模和维数

$n$-Gorenstein Projective Modules and Dimensions over Frobenius Extensions

中文关键词:  Frobenius扩张  $n$-Gorenstein投射模  $n$-Gorenstein投射维数

英文关键词:Frobenius extensions  $n$-Gorenstein projective modules  $n$-Gorenstein projective dimensions

中文摘要:

      本文讨论了Frobenius扩张上的$n$-Gorenstein投射模和$n$-Gorenstein投射维数.设$R\subset A$是环的Frobenius扩张, $M$是任意左$A$-模. 证明了: $M$是$n$-Gorenstein投射左$A$-模当且仅当$A\otimes_{R}M$和$\mathrm{Hom}_{R}(A,M)$是$n$-Gorenstein投射左$A$-模当且仅当$M$是$n$-Gorenstein投射左$R$-模.进而,当$R\subset A$是可分Frobenius扩张时,考虑了$n$-Gorenstein投射维数.

英文摘要:

      In this paper, we study $n$-Gorenstein projective modules over Frobenius extensions and $n$-Gorenstein projective dimensions over separable Frobenius extensions. Let $R\subset A$ be a Frobenius extension of rings and $M$ any left $A$-module. It is proved that $M$ is an $n$-Gorenstein projective left $A$-module if and only if $A\otimes_{R}M$ and $\mathrm{Hom}_{R}(A,M)$ are $n$-Gorenstein projective left $A$-modules if and only if $M$ is an $n$-Gorenstein projective left $R$-module. Furthermore, when $R\subset A$ is a separable Frobenius extension, $n$-Gorenstein projective dimensions are considered.