A note on additive commutator groups in certain algebras

TL;DR

The paper investigates when a unital associative algebra $A$ can be decomposed as $A = Z(A) + [A,A]$, with a focus on matrix rings over division rings, generalized quaternion algebras, semisimple finite-dimensional algebras, and twisted group algebras. It proves a main theorem giving precise conditions on a division ring $D$ and an integer $n$ under which $M_n(D) = Z(M_n(D)) + [M_n(D),M_n(D)]$, and derives numerous corollaries for several algebraic classes, including a commutativity criterion for twisted group algebras in characteristic $0$. It extends the framework to spans of images of noncommutative polynomials and to generation results for division algebras by such images, offering structural insights into how the center governs noncommutativity across diverse settings. Collectively, these results illuminate the interplay between the center and additive commutators and have implications for C*-algebras and related constructions.

Abstract

We study whether a unital associative algebra $ A $ over a field admits a decomposition of the form $A = Z(A) + [A,A]$ where $ Z(A) $ is the center of $ A $ and $ [A,A] $ denotes the additive subgroup of $A$ generated by all additive commutators of $A$. Among our main considerations are the cases in which $A$ is the matrix ring over a division ring, a generalized quaternion algebra, or a semisimple finite-dimensional algebra. We also discuss some applications that do not necessarily require the decomposition, such as the case where $ A $ is the twisted group algebra of a locally finite group over a field of characteristic zero: if all additive commutators of $A$ are central, then $ A $ must be commutative.

Theorems & Definitions (21)

• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Corollary 2.4
• Corollary 2.5
• proof
• Lemma 2.6