Reduced Gröbner bases and Macaulay-Buchberger Basis Theorem over Noetherian rings

In this paper, we extend the characterization of Z[x]/〈f〉, where f ∈ Z[x] to be a free Z-module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Gröbner basis method of computing a k-vector space basis of residue class polynomial rings over a field k (Macaulay-Buchberger Basis Theorem) to rings, i.e. A[x1, . . . , xn]/a, where a ⊆ A[x1, . . . , xn] is an ideal. We give some insights into the characterization for two special cases, when A = Z and A = k[θ1, . . . , θm]. As an application of this characterization, we show that the concept of border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free A-module.