Integer-valued polynomials on commutative rings and modules

Jesse Elliott

paper

Integer-valued polynomials on commutative rings and modules

arXiv:1608.00171

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 $R [T_{1}, \dots, T_{n}] / (T_{1} (T_{1} - r_{1}), \dots, T_{n} (T_{n} - r_{n}))$ for any commutative ring $R$ and any elements $r_{1}, \dots, r_{n}$ 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 $R$ .