Matrix division with an idempotent divisor or quotient

For given matrices F and G   over an arbitrary field FF, necessary and sufficient conditions (in terms of rank, amongst others) are presented for F to divide G, where either F itself is idempotent or one of its quotients is idempotent, or both. Formulae are also given by which these quotients can be constructed. Finally, new proofs of some known results on idempotent factorization are presented in the present context to provide another perspective on the behaviour of such factorizations.