We introduce the notions of a commutative square ring $R$ and of a quadratic
map between modules over $R$, called $R$-quadratic map. This notion generalizes
various notions of quadratic maps between algebraic objects in the literature.
We construct a category of quadratic maps between $R$-modules and show that it
is a right-quadratic category and has an internal Hom-functor. Along our way,
we recall the notions of a general square ring $R$ and of a module over $R$,
and discuss their elementary properties in some detail, adopting an operadic
point of view. In particular, it turns out that the associated graded object of
a square ring $R$ is a nilpotent operad of class 2, and the associated graded
object of an $R$-module is an algebra over this operad, in a functorial way.
This generalizes the well-known relation between groups and graded Lie algebras
(in the case of nilpotency class 2). We also generalize some elementary notions
from group theory to modules over square rings.