Macaulay-Buchberger Basis Theorem for Residue Class Polynomial Rings with Torsion and Border Bases over Rings

In this paper we generalize the Macaulay-Buchberger basis theorem to the case, where the residue class polynomial ring over a Noetherian ring is not necessarily a free module. Recently, this theorem has been extended from polynomial rings over fields to rings, when residue class polynomial ring is free in (Francis & Dukkipati, 2014). As an application of this generalization we develop a theory of border bases for ideals where the corresponding residue class rings are finitely generated and have torsion. We present a border division algorithm and prove termination of the algorithm for a special class of border bases. We show the existence of such border bases and present some characterizations in this regard. We also show that Pauer (2007) reduced Gröbner bases with respect to all possible term orders is contained in this class of border bases.