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Subnormal subgroups of almost locally simple artinian algebras with involutions

Dau Thi Hue, Huynh Viet Khanh, Bui Xuan Hai

Abstract

In this paper, we investigate subnormal subgroups of the multiplicative group of an almost locally simple artinian algebra with involution. In particular, we show that if either the set of traces or the set of norms of such a subgroup with respect to this involution is central, then the algebra must be either a quaternion division algebra or the matrix ring of degree $2$ over a field.

Subnormal subgroups of almost locally simple artinian algebras with involutions

Abstract

In this paper, we investigate subnormal subgroups of the multiplicative group of an almost locally simple artinian algebra with involution. In particular, we show that if either the set of traces or the set of norms of such a subgroup with respect to this involution is central, then the algebra must be either a quaternion division algebra or the matrix ring of degree over a field.
Paper Structure (9 sections, 22 theorems, 23 equations)

This paper contains 9 sections, 22 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Some classes of algebras
  3. Weakly locally finite division rings
  4. Almost locally simple PI algebras
  5. Locally central simple and finite dimensional algebras
  6. Almost locally simple artinian $K$-algebras
  7. Involutions on rings and their restrictions
  8. Skew linear groups with involutions
  9. Almost locally simple artinian algebras with involution

Key Result

Lemma 2.1

Let $D$ be a division ring with center $F$, and $\mathbb{P}$ the prime subfield of $D$. Then, $D$ is weakly locally finite if and only if $K(S)$ is centrally finite for every finite subset $S$ and every subfield $K$ of $D$ such that $\mathbb{P}\subseteq K\subseteq F$.

Theorems & Definitions (48)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Definition 2.7: bar-on-gilat-matzri-vishne
  • ...and 38 more