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.