关于FP-内射维数的一个注记

A Note on FP-Injective Dimension

中文关键词:  广义Gorenstein维数  $FP$-内射维数  左正交维数.

英文关键词:generalized Gorenstein dimension  $FP$-injective dimension  left orthogonal dimension.

中文摘要:

      设$R$是左凝聚环, $S$是右凝聚环,$_R\omega_S$是一个忠实平衡自正交双模. 在本文中, 我们给出了由 $l.FP-id_R(\omega)$有限推出~$G-dim_{\omega}(M)$有限的一个充分条件,其中$M\in$mod$R$. 我们不仅推广了黄和唐$^{[1]}$关于~$FP$-内射维数和广义Gorenstein维数的关系,而且还给出了在~$G-dim_{\omega}(M)$有限时,左正交维数和广义Gorenstein维数是相等的结论.

英文摘要:

      Let $R$ and $S$ be a left coherent ring and a right coherent ring respectively, $_R\omega_S$ be a faithfully balanced self-orthogonal bimodule. We give a sufficient condition to show that $l.FP\mbox{-}\id_R(\omega)<\infty$ implies $ G\mbox{-}\dim_{\omega}(M)<\infty$, where $M\in$ mod\,$R$. This result generalizes the result by Huang and Tang about the relationship between the FP-injective dimension and the generalized Gorenstein dimension in 2001. In addition, we get that the left orthogonal dimension is equal to the generalized Gorenstein dimension when $G\mbox{-}\dim_{\omega}(M)$ is finite.