The singular linear preservers of non-singular matrices

Given an arbitrary field K, we reduce the determination of the singular endomorphisms f of Mn(K) such that f(GLn(K))⊂GLn(K) to the classification of n-dimensional division algebras over K. Our method, which is based upon Dieudonné’s theorem on singular subspaces of Mn(K), also yields a proof for the classical non-singular case.