Cohen-Macaulayness of Nagata idealization (Q6635923)

Let \(R\) be a commutative ring, \(\mathfrak{a}\) an ideal of \(R\), and \(M\) a finitely generated \(R\)-module. The idealization of \(M\) is the ring \(R\ltimes M\), which is defined as an abelian group to be \(R\oplus M\), and whose ring multiplication is given by \((a_{1},x_{1})(a_{2},x_{2}) = (a_{1}a_{2}, a_{1}x_{2} + a_{2}x_{1})\) for every \(a_{1}, a_{2}\in R\) and \(x_{1}, x_{2}\in M\). The goal of this paper is to investigate the algebraic properties of \(R\ltimes M\) that are related to those of \(R\) and \(M\). Specifically, the authors provide characterizations for the Cohen-Macaulayness, sequentially Cohen-Macaulayness, generalized Cohen-Macaulayness, and graded maximal depth property of \(R\ltimes M\) with respect to \(\mathfrak{a}\ltimes M\) in terms of the corresponding properties for \(R\) and \(M\) with respect to \(\mathfrak{a}\).