## 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 = D/dD 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.

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 = D/dD 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.

Original language | English |
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Pages (from-to) | 44-69 |

Journal | Linear Algebra and its Applications |

Volume | 494 |

DOIs | |

Publication status | Published - 2016 |

## Fields of Expertise

- Sonstiges

## Treatment code (Nähere Zuordnung)

- Basic - Fundamental (Grundlagenforschung)