RESULTS ON AMALGAMATION ALONG A SEMIDUALIZING IDEAL

Abstract

Let $R$ be a commutative Noetherian ring and let $I$ be asemidualizing ideal of $R$. In this paper, it is shown that the$G_{I}$-projective, $G_{I}$-injective, and $G_{I}$-flat dimensionsagree with $Gpd _{R\bowtie I}(-)$, $Gid _{R\bowtie I}(-)$, and$Gfd _{R\bowtie I}(-)$, respectively. Also, it is proved that fora non-negative integer $n$ if $\sup \{\mathcal{GP}_{I}-pd _{R}(M) \midM\in \mathcal{M}(R) \}\leq n$ (or $\sup \{\mathcal{GI}_{I}-id_{R}(M) \mid M\in \mathcal{M}(R) \}\leq n)$, then for every projective$(R\bowtie I)$-module $P$ we have $id _{R\bowtie I}(P)\leq n$, andfor every injective $(R\bowtie I)$-module $E$ we have $pd_{R\bowtie I}(E)\leq n$.