  We present an adapted construction of algebraic circuits over the reals
introduced by Cucker and Meer to arbitrary infinite integral domains and
generalize the $\mathrm{AC}_{\mathbb{R}}$ and
$\mathrm{NC}_{\mathbb{R}}$-classes for this setting. We give a theorem in the
style of Immerman's theorem which shows that for these adapted formalisms, sets
decided by circuits of constant depth and polynomial size are the same as sets
definable by a suitable adaptation of first-order logic. Additionally, we
discuss a generalization of the guarded predicative logic by Durand, Haak and
Vollmer and we show characterizations for the $\mathrm{AC}_{R}$ and
$\mathrm{NC}_{R}$ hierarchy. Those generalizations apply to the Boolean
$\mathrm{AC}$ and $\mathrm{NC}$ hierarchies as well. Furthermore, we introduce
a formalism to be able to compare some of the aforementioned complexity classes
with different underlying integral domains.
