Reduced Gr\"obner Bases and Macaulay-Buchberger Basis Theorem over Noetherian Rings

In this paper, we extend the characterization of $\mathbb{Z}[x]/\ < f \ >$, where $f \in \mathbb{Z}[x]$ to be a free $\mathbb{Z}$-module to multivariate polynomial rings over any commutative Noetherian ring, $A$. The characterization allows us to extend the Gr\"obner basis method of computing a $\Bbbk$-vector space basis of residue class polynomial rings over a field $\Bbbk$ (Macaulay-Buchberger Basis Theorem) to rings, i.e. $A[x_1,\ldots,x_n]/\mathfrak{a}$, where $\mathfrak{a} \subseteq A[x_1,\ldots,x_n]$ is an ideal. We give some insights into the characterization for two special cases, when $A = \mathbb{Z}$ and $A = \Bbbk[\theta_1,\ldots,\theta_m]$. 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.