On bilinear maps on matrices with applications to commutativity preservers

Let Mn 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 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 Mn into Mn for the case where C is an arbitrary field; moreover, this description is valid in every finite-dimensional central simple algebra.