Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces

Given a polynomial matrix B(s), we consider the class of nonsingular polynomial matrices L(s) such that B(s) = R(s)L(s) for some polynomial matrix R(s). It is shown that finding such factorizations is equivalent to finding (A,B)-invariant subspaces in the kernel of C where A,B,C are linear maps determined by B(s). In particular, the results yield, as a corollary, a method to determine simultaneously a row proper greatest right divisor of a left invertible polynomial matrix as well as the resulting polynomial matrix whose greatest right divisors are unimodular. The results also relate, the same way, such subspaces of constant systems (C,A,B) where (C,A) is observable and (A,B) is reachable, to the nonsingular right factors of the numerator polynomial matrices in coprime factorizations of the form D^-1 (s)B(s) of their transfer matrices.