Linear operators that preserve zero-term rank over fields and rings (Q5957189)

Let \(M_{m,n}(\mathbb{F)}\) denote the space of all \(m\times n\) matrices over a field \(\mathbb{F}\). The authors define the ``zero-term rank'' \(z(A)\) of \(A\in M_{m,n}(\mathbb{F)}\) to be the maximum number of zeros on a generalized diagonal of \(A\). They generalize results of a previous paper by \textit{L. B. Beasley, S.-Z. Song} and \textit{S.-G. Lee} [Linear Multilinear Algebra 48, No. 4, 313-318 (2001; Zbl 0987.15002)] to characterize the linear transformations \(T\) of \(M_{m,n}(\mathbb{F)}\) into itself such that \(z(TA)=z(A)\) for all \(A\) (respectively, for all \(A\) with \(z(A)=1\)) in the cases where \(\left|\mathbb{F}\right|\geq mn+2\) or \(char(\mathbb{F})=2\).