  We prove that, in any field of characteristic not two and not three except
the five-element field, each element decomposes into a product of four factors
whose sum vanishes. We also find all $k,n,q$ such that every $n\times n$ matrix
over the $q$-element field decomposes into a product of $k$ commuting matrices
whose sum vanishes.

Rings and Algebras;Commutative Algebra;History and Overview;Number Theory