Ray unit ($U_{\mathfrak{c}}={\mathcal{O}}_K^{\times }\cap K_{\mathfrak{c},1}^{\times }$)

A ray unit for a modulus of a number field is a unit congruent to one modulo every finite prime power in the modulus and positive at every real place in it. The group of ray units is the kernel of the unit group map to the residue and signature group of a modulus.