On Polynomial Rings Over Nil Rings in Several Variables and the Central Closure of Prime Nil Rings

TL;DR

This paper proves that polynomial rings over nil rings are radical and cannot map onto rings with identity, and that the central closure of prime nil rings cannot be simple with identity, addressing open questions in ring theory.

Contribution

It establishes new properties of polynomial rings over nil rings and the structure of prime nil rings, solving previously open problems in algebra.

Findings

Polynomial rings over nil rings are Brown-McCoy radical.

The central closure of a prime nil ring cannot be a simple ring with identity.

Answers to open questions by Puczylowski, Smoktunowicz, and Beidar.

Abstract

We prove that the ring of polynomials in several commuting indeterminates over a nil ring cannot be homomorphically mapped onto a ring with identity, i.e. it is Brown-McCoy radical. It answers a question posed by Puczylowski and Smoktunowicz. We also show that the central closure of a prime nil ring cannot be a simple ring with identity solving a problem due to Beidar.