Grr Obner Bases for Polynomial Ideals over Commutative Noetherian Rings and Related Results ?

We present a completion-like procedure for constructing weak and strong Grr obner bases for polynomial ideals over commutative Noetherian rings with unit. This generalizes all the known algorithms for computing Grr obner bases for polynomial ideals over various diierent coeecient domains. The coeecient domain is incorporated using constraints. Constraints allow us to describe an optimized procedure for computing Grr obner bases. The optimization restricts the number of su-perpositions that need to be considered. Weak Grr obner bases are shown to extend to Grr obner bases under an additional ordering assumption on coeecient domain. The conditions on the coeecient domain are the weakest possible and are shown to carry over from ring B to BX ], thus giving a hierarchic algorithm for construction of Grr obner bases.