Cohen–Macaulayness of Nagata idealization

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text], and [Formula: see text] a finitely generated [Formula: see text]-module. We consider the idealization [Formula: see text] of [Formula: see text] over [Formula: see text]. The goal of this paper is to investigate algebraic properties of [Formula: see text] that are related to those of [Formula: see text] and [Formula: see text]. Specifically, we provide characterizations for the Cohen–Macaulayness, sequentially Cohen–Macaulayness, generalized Cohen–Macaulayness, and graded maximal depth property of [Formula: see text] with respect to [Formula: see text], in terms of the corresponding properties for [Formula: see text] and [Formula: see text] with respect to [Formula: see text].