  The main goal of this paper is to show that if a real valued function defined
on a groupoid satisfies a certain Levi--Civita-type functional equation, then
it also fulfills a Cauchy--Schwarz-type functional inequality. In particular,
if the groupoid is the multiplicative structure of commutative ring, then we
can establish the existence of nontrivial additive functions satisfying
inequalities connected to the multiplicative structure.
