Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mikhail Chebotar, Wen-Fong Ke, Pjek-Hwee Lee, Edmund R. Puczylowski

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.

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

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.

Paper Structure

This paper contains 3 sections, 8 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Simple Facts from Convex Geometry
  3. Proof of the Main Theorem

Key Result

Theorem 1

Let $R$ be a prime nil ring with extended centroid $C$. Let $c_{1},\ldots,c_{n}\in C$, $n\ge 1$, be such that $S=R[c_{1},\ldots,c_{n}]$ is a simple ring. Then $S$ has a zero center.

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • ...and 3 more