Nonsurjective zero product preservers between matrix spaces over an arbitrary field

A map $\mathrm{\Phi }$ between matrices is said to be zero product preserving if $$\mathrm{\Phi }(A)\mathrm{\Phi }(B)=0\text{whenever}AB=0.$$ In this paper, we give concrete descriptions of an additive/linear zero product preserver $\Phi: {\bf M}_n($\mathbb{F}$) \rightarrow {\bf M}_r($\mathbb{F}$)$ between matrix algebras of different dimensions over an arbitrary field $\mathbb{F}$. In particular, we show that if $\mathrm{\Phi }$ is linear and preserves zero products then $$\mathrm{\Phi }(A)=S\left(\begin{array}{cc}{R}_1\otimes A& 0\\ 0& {\mathrm{\Phi }}_0(A)\end{array}\right)S^{-1},$$ for some invertible matrices $R_1$ in ${\mathbf{M}}_k(\mathbb{F})$, $S$ in ${\bf M}_r($\mathbb{F}$)$andazeroproductpreservinglinearmap$\Phi_0: {\bf M}n($\mathbb{F}$) \rightarrow {\bf M}{r-nk}($\mathbb{F}$)$ into nilpotent matrices. If $\mathrm{\Phi }(I_n)$ is invertible, then ${\mathrm{\Phi }}_0$ is vacuous. In general, the structure of ${\mathrm{\Phi }}_0$ could be quite arbitrary, especially when ${\mathrm{\Phi }}_0({\mathbf{M}}_n(\mathbb{F}))$ has trivial multiplication, i.e., ${\mathrm{\Phi }}_0(X){\mathrm{\Phi }}_0(Y)=0$ for all $X,Y$ in ${\mathbf{M}}_n(\mathbb{F})$. We show that if ${\mathrm{\Phi }}_0(I_n)=0$ or $r-nk\le n+1$, then $\Phi_0({\bf M}_n($\mathbb{F}$))$ indeed has trivial multiplication. More generally, we characterize subspaces $\mathbf{V}$ of square matrices satisfying $XY=0$ for any $X,Y\in \mathbf{V}$. Similar results for double zero product preserving maps are obtained.