Recently, Cuntz and Li introduced the
C
∗
C^*
-algebra
A
[
R
]
\mathfrak {A}[R]
associated to an integral domain
R
R
with finite quotients. In this paper, we show that
A
[
R
]
\mathfrak {A}[R]
is a partial group algebra of the group
K
⋊
K
×
K\rtimes K^\times
with suitable relations, where
K
K
is the field of fractions of
R
R
. We identify the spectrum of these relations and we show that it is homeomorphic to the profinite completion of
R
R
. By using partial crossed product theory, we reconstruct some results proved by Cuntz and Li. Among them, we prove that
A
[
R
]
\mathfrak {A}[R]
is simple by showing that the action is topologically free and minimal.