Let
K
K
be a number field with ring of integers
R
R
. Given a modulus
m
\mathfrak {m}
for
K
K
and a group
Γ
\Gamma
of residues modulo
m
\mathfrak {m}
, we consider the semidirect product
R
⋊
R
m
,
Γ
R\rtimes R_{\mathfrak {m},\Gamma }
obtained by restricting the multiplicative part of the full
a
x
+
b
ax+b
-semigroup over
R
R
to those algebraic integers whose residue modulo
m
\mathfrak {m}
lies in
Γ
\Gamma
, and we study the left regular C*-algebra of this semigroup. We give two presentations of this C*-algebra and realize it as a full corner in a crossed product C*-algebra. We also establish a faithfulness criterion for representations in terms of projections associated with ideal classes in a quotient of the ray class group modulo
m
\mathfrak {m}
, and we explicitly describe the primitive ideals using relations only involving the range projections of the generating isometries; this leads to an explicit description of the boundary quotient. Our results generalize and strengthen those of Cuntz, Deninger, and Laca and of Echterhoff and Laca for the C*-algebra of the full
a
x
+
b
ax+b
-semigroup. We conclude by showing that our construction is functorial in the appropriate sense; in particular, we prove that the left regular C*-algebra of
R
⋊
R
m
,
Γ
R\rtimes R_{\mathfrak {m},\Gamma }
embeds canonically in the left regular C*-algebra of the full
a
x
+
b
ax+b
-semigroup. Our methods rely heavily on Li’s theory of semigroup C*-algebras.