On bilinear maps on matrices with applications to commutativity preservers

Abstract

Let M n be the algebra of all n × n matrices over a commutative unital ring C , and let L be a C -module. Various characterizations of bilinear maps { . , . } : M n × M n → L with the property that { x , y } = 0 whenever x any y commute are given. As the main application of this result we obtain the definitive solution of the problem of describing (not necessarily bijective) commutativity preserving linear maps from M n into M n for the case where C is an arbitrary field; moreover, this description is valid in every finite-dimensional central simple algebra.