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

On Ring Class Fields of Number Rings