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

Let (R ,) be a Noetherian local ring and M a finitely generated R -module. In this paper, we study the polynomial type and the sequential polynomial type of the idealization R ⋉ M of R and M 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. [ABSTRACT FROM AUTHOR]