2025/05/19 by Nguyen Thi Thai Ha, Ha, Nguyen Thi Thai, Tran Nam Son +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2505.13303
openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.