- Let M be the box spline associated with an sn matrix . If has only integer entries, then the phenomenon of linear independence of the integer translates of M is now well understood. When the translates of M are restricted to submodules of Zs, the corresponding problem has received some attention recently. This problem is intimately connected with the linear independence of integer translates of box splines associated with rational matrices. In this paper, a sufficient condition for the linear independence of such translates is given. This result extends the known theorem for integer matrices. Some inherent differences between the two cases are highlighted. These differences indicate that an exact necessary and sufficient condition for independence in the case of rational matrices may be difficult to obtain. However, a complete characterization is provided for the univariate case. 1990.