On the splitting ring of a polynomial

TL;DR

This paper investigates the algebraic structure of the splitting ring of a polynomial over a possibly non-commutative ring, focusing on the ideal generated by elementary symmetric polynomials and the resulting quotient ring.

Contribution

It provides a detailed analysis of the ideal and quotient ring associated with the splitting ring of a polynomial over a non-commutative ring.

Findings

Characterization of the ideal $I_f$ in the polynomial ring

Description of the quotient ring structure

Extension of classical results to non-commutative rings

Abstract

Let $f(Z)=Z^n-a_{1}Z^{n-1}+\cdots+(-1)^{n-1}a_{n-1}Z+(-1)^na_n$ be a monic polynomial with coefficients in a ring~$R$ with identity, not necessarily commutative. We study the ideal $I_f$ of $R[X_1,\dots,X_n]$ generated by $\sigma_i(X_1,\dots,X_n)-a_{i}$, where $\sigma_1,\dots,\sigma_n$ are the elementary symmetric polynomials, as well as the quotient ring $R[X_1,\dots,X_n]/I_f$.