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We focus on Grobner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the “predictable leading monomial (PLM) property” that is shared by minimal Grobner bases of modules in F [ x ] q , no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property (= row reducedness), a terminology introduced by Forney in the 70’s. Because of the presence of zero divisors, minimal Grobner bases over a finite ring of the type Z p r (where p is a prime integer and r is an integer > 1 ) do not necessarily have the PLM property. In this paper we show how to derive, from an ordered minimal Grobner basis, a so-called “minimal Grobner p-basis” that does have a PLM property. We demonstrate that minimal Grobner p-bases lend themselves particularly well to derive minimal realization parametrizations over Z p r . Applications are in coding and sequences over Z p r .