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 $Gp{d}_{R\bowtie I}(-)$, $Gi{d}_{R\bowtie I}(-)$, and$Gf{d}_{R\bowtie I}(-)$, respectively. Also, it is proved that fora non-negative integer $n$ if $sup\{{\mathcal{GP}}_I-p{d}_R(M)\text{midM}\in \mathcal{M}(R)\}\le n$ (or $sup\{{\mathcal{GI}}_I-i{d}_R(M)\mid M\in \mathcal{M}(R)\}\le n)$, then for every projective$(R\bowtie I)$-module $P$ we have $i{d}_{R\bowtie I}(P)\le n$, andfor every injective $(R\bowtie I)$-module $E$ we have $p{d}_{R\bowtie I}(E)\le n$.