Square-zero factorization of matrices

For given m×nm×n matrices G and F   over an arbitrary field FF, necessary and sufficient conditions (in terms of rank, amongst others) are presented for F to divide G with a square-zero quotient. These results are then used to extend the results of Novak [3] on square-zero factorization to matrices over an arbitrary field. The ranks that the square-zero factors can have are also investigated. Formulae are also presented by which these quotients can be constructed when this type of division is possible.