  We focus on Gr\"obner 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 Gr\"{o}bner 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
Gr\"{o}bner 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 Gr\"{o}bner basis, a
so-called "minimal Gr\"{o}bner p-basis" that does have a PLM property. We
demonstrate that minimal Gr\"obner 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.
