  This paper defines a generalized notion of "ray class group" associated to an
arbitrary order in a number field together with an arbitrary ray class modulus
for that order. It shows existence of ray class fields corresponding to these
ray class groups. These ray class groups (resp., ray class fields) specialize
to classical ray class groups (resp., fields) of a number field in the case of
the maximal order, and they specialize to ring class groups (resp., fields) of
orders in the case of trivial modulus. The paper gives an exact sequence for
simultaneous change of order and change of modulus. As a consequence, we
identify the ray class field of an order with a given modulus as a specific
subfield of a ray class field of the maximal order with a larger modulus. We
also uniquely identify each ray class field of an order in terms of the
splitting behavior of primes. An appendix extends some structural results of
the paper to ray class monoids of the order (which include non-invertible
elements).
