On \((n,d)\)-property in amalgamated algebra (Q2806095)

Throughout this review, \(R\) stands for a commutative ring with identity. An \(R\)-module \(M\) is said to be an \(n\)-presented module provided there exists an exact sequence, NEWLINE\[NEWLINE F_n\rightarrow F_{n-1}\rightarrow \cdots \rightarrow F_1\rightarrow F_0\rightarrow M\rightarrow 0, NEWLINE\]NEWLINE consisting of finitely generated free \(R\)-modules \(F_0,\ldots, F_n\).NEWLINENEWLINEFor an ordered pair \((n,d)\) of non-negative integers \(n,d\), a ring \(R\) is said to be an \((n,d)\)-ring provided any \(n\)-presented \(R\)-module \(M\) has projective dimension at most \(d\) (over \(R\)). The notion of \((n,d)\)-rings was introduced by \textit{D. L. Costa} in [Commun. Algebra 22, No. 10, 3997--4011 (1994; Zbl 0814.13010)]. For instance, any \((n,0)\)-domain is a field, any \((0,1)\)-domain is a Dedekind domain and any \((1,1)\)-domain is a Prüfer domain.NEWLINENEWLINEIn the paper under review, motivated by a question proposed by Costa, for certain small values of \(d\), the authors give a new class of \((2,d)\)-rings (respectively, \((3,d)\)-rings) which are neither \((2,d-1)\)-ring nor \((1,d)\)-ring (respectively, neither \((3,d-1)\)-ring nor \((2,d)\)-ring). The construction of examples involves the Nagata's idealization (in some instances), the amalgamation \(A\bowtie^f J\) with respect to certain ring homomorphism \(f:A\rightarrow B\) along an ideal \(J\) of \(B\), as well as the product of two rings \(R\) and \(S\) such that \(R\) is such an amalgamation construction and \(S\) is certain polynomial ring over a field (or over \(\mathbb{Z}\)).