A note on additive commutator groups in certain algebras

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.