Integer-valued polynomials on commutative rings and modules

TL;DR

This paper explores the extension of integer-valued polynomial theory from integral domains to more general commutative rings and modules, providing new examples and motivating further research in this broader context.

Contribution

It introduces the study of integer-valued polynomials on commutative rings and modules, with explicit computations for specific algebraic structures.

Findings

Computed integer-valued polynomials over certain quotient rings

Analyzed integer-valued polynomials over Nagata idealizations

Extended classical theory to new algebraic contexts

Abstract

The ring of integer-valued polynomials on an arbitrary integral domain is well-studied. In this paper we initiate and provide motivation for the study of integer-valued polynomials on commutative rings and modules. Several examples are computed, including the integer-valued polynomials over the ring $R[T_1,\ldots, T_n]/(T_1(T_1-r_1), \ldots, T_n(T_n-r_n))$ for any commutative ring $R$ and any elements $r_1, \ldots, r_n$ of $R$, as well as the integer-valued polynomials over the Nagata idealization $R(+)M$ of $M$ over $R$, where $M$ is an $R$-module such that every non-zerodivisor on $M$ is a non-zerodivisor of $R$.