Nonsurjective zero product preservers between matrix spaces over an arbitrary field

Abstract

A map Φ between matrices is said to be zero product preserving if In this paper, we give concrete descriptions of an additive/linear zero product preserver between matrix algebras of different dimensions over an arbitrary field , and . In particular, we show that if Φ is linear and preserves zero products then for some invertible matrices in , S in and a zero product preserving linear map into nilpotent matrices. If is invertible, then is vacuous. In general, the structure of could be quite arbitrary, especially when has trivial multiplication, i.e. for all X, Y in . We show that if or , then indeed has trivial multiplication. More generally, we characterize subspaces of square matrices satisfying XY = 0 for any . Similar results for double zero product preserving maps are obtained.