Abstract
Given a square matrix A with entries in a commutative ring S, the ideal of S[X] consisting of polynomials f with 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=DdD is the residue class ring of a principal ideal domain D modulo d∈D. After that we discuss two applications. We compute a decomposition of the S-module 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,Mn(D)) of all integer-valued polynomials on A.
Highlights
Matrices with entries in commutative rings arise in numerous contexts, both in pure and applied mathematics
For a general introduction to matrix theory over commutative rings we refer to the textbook of Brown [4]
The purpose of this paper is to provide a better understanding of null ideals of square matrices over residue class rings of principal ideal domains
Summary
Matrices with entries in commutative rings arise in numerous contexts, both in pure and applied mathematics. Many of the well-known results of classical linear algebra do not hold in this general setting This is the case even if the underlying ring is a domain (but not a field). As a finitely generated module over a principal ideal ring, D/plD[A] decomposes into a direct sum of cyclic submodules with uniquely determined invariant factors, according to [4, Theorem 15.33] We describe this decomposition explicitly and find a strong relationship to the generating set of ND/plD(A) from Section 2. We give an explicit description of the ring Int(A, Mn(D)) using the generating set of the null ideal of A modulo finitely many prime powers pl Once this description is given, the ring Int-Im(A, Mn(D)) of images of A under integer-valued polynomials is determined
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