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On nonmatrix varieties of associative rings

Thiago Castilho de Mello, Felipe Yukihide Yasumura

Abstract

We study nonmatrix varieties of $\mathbf{k}$-algebras, where $\mathbf{k}$ is a unital commutative ring. Our results extend to this generality known results for the case in which $\mathbf{k}$ is an infinite field. Also, we generalize these results to varieties of $\mathbf{k}$-algebras not containing the algebra of $n\times n$ matrices.

On nonmatrix varieties of associative rings

Abstract

We study nonmatrix varieties of -algebras, where is a unital commutative ring. Our results extend to this generality known results for the case in which is an infinite field. Also, we generalize these results to varieties of -algebras not containing the algebra of matrices.
Paper Structure (5 sections, 19 theorems, 39 equations)

This paper contains 5 sections, 19 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonmatrix variety over rings
  4. Nonmatricial radical
  5. On varieties not containing matrix algebras of higher order

Key Result

Lemma 1

Let $\mathscr{V}$ be a variety of associative $\mathbf{k}$-algebras satisfying a polynomial identity, where $\mathbf{k}$ is a unital commutative ring, and let $\mathcal{A}\in\mathscr{V}$ be a primitive algebra. Then either $\mathcal{A}$ is a field, or $\mathscr{V}$ contains $\mathrm{M}_2(S^{-1}\bar{

Theorems & Definitions (51)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: AGPR
  • Lemma 4: KBR
  • Theorem 5
  • proof
  • Remark 6
  • Corollary 7
  • ...and 41 more