  The theory of border bases for zero-dimensional ideals has attracted several
researchers in symbolic computation due to their numerical stability and
mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014),
one can extend the concept of border bases over Noetherian rings whenever the
corresponding residue class ring is finitely generated and free. In this paper
we address the following problem: Can the concept of border basis over
Noetherian rings exists for ideals when the corresponding residue class rings
are finitely generated but need not necessarily be free modules? We present a
border division algorithm and prove the termination of the algorithm for a
special class of border bases. We show the existence of such border bases over
Noetherian rings and present some characterizations in this regard. We also
show that certain reduced Gr\"{o}bner bases over Noetherian rings are contained
in this class of border bases.
