Integer-valued polynomials on commutative rings and modules

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 $R[T_1,\dots ,T_n]∕\left(T_1\right(T_1-r_1),\dots ,T_n(T_n-r_n\left)\right)$ 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. These examples are motivated by and provide motivation for the study of integer-valued derivatives.

KEYWORDS:

• Dual numbers
• integer-valued polynomial
• module
• ring

MATHEMATICS SUBJECT CLASSIFICATION:

• 13F20
• 13A99