On G-\((n,d)\)-rings
Let \(R\) denote a commutative ring. An \(R\)-module \(M\) is called \(n\)-presented whenever there is an exact sequence \(F_n \to \ldots \to F_0 \to M \to 0\) of finitely generated free \(R\)-modules \(F_i, i = 0,\ldots, n\). In his paper [Commun. Algebra 22, No. 10, 3997--4011 (1994; Zbl 0814.13010)] \textit{D. L. Costa} studied the following doubly filtered set of classes of rings: For non-negative integers \(n, d\) a commutative ring \(R\) is called \((n,d)\)-ring if the projective dimension satisfies \(\text{pd}_R M \leq d\) for each \(n\)-presented \(R\)-module \(M\). The authors extend Costa's idea in the following way: A commutative ring \(R\) is called G-\((n,d)\)-ring provided every \(n\)-presented \(R\)-module \(M\) has Gorenstein projective dimension \(\text{Gpd}_R M \leq d\), where Gorenstein projective dimension is introduced by \textit{E. E. Enochs} and \textit{O. M. G. Jenda} [Math. Z. 220, No. 4, 611--633 (1995; Zbl 0845.16005)]. The authors characterize \(n\)-coherent G-\((n,0)\)-rings and give various examples of G-\((n,d)\)-rings.
scientific article; zbMATH DE number 6066671
scientific article