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

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