Skew prime polynomial matrices

A pair of polynomial matrices, P(s) and Q(s) , is defined to be "externally skew prime" if and only if a solution, M(s), N(s) , to the polynomial matrix equation P(s)M(s)+N(s)Q(s)=I exists. It is shown that P(s) and Q(s) are externally skew prime if and only if Q(s)P(s)= \bar{P}(s)R(s) with Q(s) and \bar{P}(s) relatively left prime and P(s) and R(s) relatively right prime. This observation implies a new constructive procedure for determining M(s) and N(s) where P(s) and Q(s) are found to be externally skew prime and P(s) is nonsingular. A new procedure for obtaining solutions to the more general polynomial matrix equation, P(s)M(s)+N(s)Q(s)= V(s) , based on the notion of skew-prime polynomial matrices is also presented. A characterization of all solutions when V(s)= I is also given, under appropriate assumptions, and then employed to determine a unique solution to this polynomial matrix equation.