Inclusion among commutators of elementary subgroups

TL;DR

This paper advances the understanding of elementary commutator subgroups in algebraic rings by providing explicit congruences and detailed descriptions of their structure, including lattice formations and subgroup relations.

Contribution

It introduces an explicit triple congruence for elementary commutators and refines classical results, offering new insights into the structure of commutator subgroups in algebraic rings.

Findings

Explicit triple congruence for elementary commutators

Complete description of lattice of commutator subgroups

New inclusions among elementary commutator subgroups

Abstract

In the present paper we continue the study of the elementary commutator subgroups $[E(n,A),E(n,B)]$, where $A$ and $B$ are two-sided ideals of an associative ring $R$, $n\ge 3$. First, we refine and expand a number of the auxiliary results, both classical ones, due to Bass, Stein, Mason, Stothers, Tits, Vaserstein, van der Kallen, Stepanov, as also some of the intermediate results in our joint works with Hazrat, and our own recent papers [40,41]. The gimmick of the present paper is an explicit triple congruence for elementary commutators $[t_{ij}(ab),t_{ji}(c)]$, where $a,b,c$ belong to three ideals $A,B,C$ of $R$. In particular, it provides a sharper counterpart of the three subgroups lemma at the level of ideals. We derive some further striking corollaries thereof, such as a complete description of generic lattice of commutator subgroups $[E(n,I^r),E(n,I^s)]$, new inclusions among multiple elementary commutator subgroups, etc.