A new set of necessary and sufficient conditions for a system to be prime

In this note, we first generalize the notion of the interactor (Wolovich and Falb [1]), and modify the input-output structure algorithm of Silverman [2], in order to obtain simple relations between the dimension of supremal output nulling invariant subspace V
^*
and the degree of the determinant of the interactor
\partial_{T}
and the integers obtained from the structure algorithm. Then, we show that these relations can be used to derive a new set of necessary and sufficient conditions for a linear multivariable system to be prime (Morse [3]).