On the uniform bound of reducibility index of parameter ideals of idealizations

Abstract

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] a finitely generated [Formula: see text]-module. In this paper, we study the polynomial type and the sequential polynomial type of the idealization [Formula: see text] of [Formula: see text] and [Formula: see text] introduced by Cuong [On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings, Nagoya Math. J. 125 (1992) 105–114] and Nhan-Dung-Chau [A measure for non-sequential Cohen Macaulayness of finitely generated modules, J. Algebra 468 (2016) 275–295], respectively. In addition, we give a uniform bound for the reducibility index of parameter ideals of idealizations whose polynomial type and sequential polynomial type are at most one.