王淼,王占平.Frobenius扩张上的$n$-Gorenstein投射模和维数[J].数学研究及应用,2021,41(1):25~32 |
Frobenius扩张上的$n$-Gorenstein投射模和维数 |
$n$-Gorenstein Projective Modules and Dimensions over Frobenius Extensions |
投稿时间:2020-02-16 修订日期:2020-09-26 |
DOI:10.3770/j.issn:2095-2651.2021.01.004 |
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 |
基金项目:国家自然科学基金(Grant No.11561061). |
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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. |
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