Polynomial functions on non-commutative rings - a link between ringsets and null-ideal sets

TL;DR

This paper explores the relationship between polynomial functions on subsets of non-commutative rings, focusing on conditions under which certain polynomial sets form rings or ideals, revealing structural links in non-commutative algebra.

Contribution

It establishes connections between sets where integer-valued polynomials form a ring and sets where zero polynomials form an ideal in non-commutative rings.

Findings

Sets with integer-valued polynomials forming a ring are characterized.

Sets where zero polynomials form an ideal are identified.

Links between these two types of sets are demonstrated.

Abstract

Regarding polynomial functions on a subset $S$ of a non-commutative ring $R$, that is, functions induced by polynomials in $R[x]$ (whose variable commutes with the coefficients), we show connections between, on one hand, sets $S$ such that the integer-valued polynomials on $S$ form a ring, and, on the other hand, sets $S$ such that the set of polynomials in $R[x]$ that are zero on $S$ is an ideal of $R[x]$.