Primeness of multivariate Laurent polynomial matrices

Different characterizations of primeness of Laurent polynomial matrices that are mutually equivalent in the case of univariate polynomials lead to various primeness concepts in the multivariate situation. Applied to systems theory, this fact gives rise to a more refined view to observability and controllability of multidimensional discrete systems defined over Z
^r
. Algorithms for testing primeness properties are given based on computer algebraic techniques.