Title: On rings with finite self-injective dimension Ⅱ
Abstract:For a module $M$ over a ring $R$ (with an identity), pd $(M)$ and id $(M)$ denote the projective and injective dimension of $M$ , respectively.In the previous paper [5] and [6], we called a (left and ...For a module $M$ over a ring $R$ (with an identity), pd $(M)$ and id $(M)$ denote the projective and injective dimension of $M$ , respectively.In the previous paper [5] and [6], we called a (left and right) noether ring $R$ n-Gorenstein if id $(_{R}R)\leqq n$ and id $(R_{R})\leqq n$ for an $n\geqq 0$ , and Gorenstein if $R$ is n-Gorenstein for some $n$ .This note is concerned with two subjects on Gorenstein rings.In \S 1, we consider the modules of finite projective or injective dimenSion over a Gorenstein ring and, first, show that the finiteness of projective dimension coincides with one of injective dimension.Then it follows that the highest finite projective (or injec- tive) dimension is $n$ for modules over an n-Gorenstein ring and, next, such modules over an artinian Gorenstein ring are investigated.Finally, we present some example to compare with Auslander's definition of an n-Gorenstein ring.In \S 2, for a Gorenstein ring $R$ , we consider a quasi-Frobenius extension of $R$ and show it also is a Gorenstein ring.Further we generalize [3, Corollary 8 and 8'] to the case of a quasi-Frobenius extension.Also an example concerning with a maximal quotient ring of a Gorenstein ring is presented. Modules of finite projective or injective dimensionWe start with the next proposition which states [4, Korollar 1.12] and [7, Corollary 5] more precisely: PROPOSITION 1.For a noether ring $R$ , id $(R_{R})=\sup$ {flat $\dim(E);RE$ is an injective left R-module.}.PROOF.By [2, Chap.VI, Proposition 5.3], $(^{*})$ $Tor_{i}^{R}(A_{R,R}E)\cong Hom_{R}(Ext_{R}^{i}(A_{R,R}R_{R}), RE)$ for any finitely generated right R-module $A_{R}$ , injective left R-module $RE$ and $i>0$ .First assume id $(R_{R})=n<\infty$ , then $Ext_{R}^{n+1}(A, R)=0$ for any finitely generatedRead More