  For a number field $K$, we extend the notion of the ring class field of an
order in $K$ [C. Lv and Y. Deng, SciChina. Math., 2015] to that of an arbitrary
number ring in $K$. We give both ideal-theoretic and idele-theoretic
description of this number ring class field, and characterize it as a subfield
of the ring class field of some order. As an application, we use it to give a
criterion of the solvability of a higher degree norm form equation over a
number ring and finally describe algorithms to compute this field.
