Null ideals of matrices over residue class rings of principal ideal domains

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• 作者：Roswitha Rissner ^{rissner@math.tugraz.at" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11C08 ; 11C20 ; 13F20 ; 15A15 ; 15B33 ; 15B36
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：494
• 期：Complete
• 页码：44-69
• 全文大小：484 K

文摘

Given a square matrix A with entries in a commutative ring S  , the ideal of S[X]$S[X]$ consisting of polynomials f   with f(A)=0$f(A)=0$ is called the null ideal of A  . Very little is known about null ideals of matrices over general commutative rings. First, we determine a certain generating set of the null ideal of a matrix in case $S=\raisebox{1ex}{$D$}\!\left/ \!\raisebox{-1ex}{$dD$}\right.$ is the residue class ring of a principal ideal domain D   modulo d∈D$d\in D$. After that we discuss two applications. We compute a decomposition of the S  -module S[A]$S[A]$ into cyclic S-modules and explain the strong relationship between this decomposition and the determined generating set of the null ideal of A  . And finally, we give a rather explicit description of the ring Int(A,M_n(D))$\mathrm{Int}(A,{\mathrm{M}}_n(D))$ of all integer-valued polynomials on A.