A Criterion for Annihilating Ideals of Linear Recurring Sequences over Galois Rings

Let R be a local Artin principal ideal ring, R[x] the polynomial ring over R with indeterminate x. Let π be an element of R such that <π> is the unique maximal ideal of R. Let I be a zero-dimensional ideal of R[x], and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} the radical ideal of I. In this paper we show that I is the annihilating ideal of a linear recurring sequence over R if and only if I satisfies the following formula\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}