  In this paper, we generalize the notion of border bases of zero-dimensional
polynomial ideals to the module setting. To this end, we introduce order
modules as a generalization of order ideals and module border bases of
submodules with finite codimension in a free module as a generalization of
border bases of zero-dimensional ideals in the first part of this paper. In
particular, we extend the division algorithm for border bases to the module
setting, show the existence and uniqueness of module border bases, and
characterize module border bases analogously like border bases via the special
generation property, border form modules, rewrite rules, commuting matrices,
and liftings of border syzygies. Furthermore, we deduce Buchberger's Criterion
for Module Border Bases and give an algorithm for the computation of module
border bases that uses linear algebra techniques. In the second part, we
further generalize the notion of module border bases to quotient modules. We
then show the connection between quotient module border bases and special
module border bases and deduce characterizations similar to the ones for module
border bases. Moreover, we give an algorithm for the computation of quotient
module border bases using linear algebra techniques, again. At last, we prove
that subideal border bases are isomorphic to special quotient module border
bases. This isomorphy immediately yields characterizations and an algorithm for
the computation of subideal border bases.
