ACCR ring of the forms $\mathcal {A}[X]$ and $\mathcal {A}[[X]]$

Abstract

Let $\mathcal {A}=(A_n)_{n\in \mathbb {N}}$ be an ascending chain of commutative rings with identity and let $\mathcal {A}[X]$ (respectively, $\mathcal {A}[[X]]$) be the ring of polynomials (respectively, power series) with coefficient of degree n in $A_n$ for each $n\in \mathbb {N}$ (Hamed and Hizem in Commun Algebra 43:3848–3856, 2015; Haouat in Thèse de doctorat. Faculté des Sciences de Tunis, 1988). An A-module M is said to satisfy ACCR if the ascending chain of residuals of the form $N:B\subseteq N:B^2\subseteq N:B^3\subseteq \cdots$ terminates for every submodule N of M and for every finitely generated ideal B of A (Lu in Proc Am Math Soc 117:5–10, 1993). We give necessary and sufficient condition for the ring $\mathcal {A}[X]$ (respectively, $\mathcal {A}[[X]]$) to satisfy ACCR.