  We introduce a notion of $R$-quadratic maps between modules over a
commutative ring $R$ which generalizes several classical notions arising in
linear algebra and group theory. On a given module $M$ such maps are
represented by $R$-linear maps on a certain module $P^2_R(M)$. The structure of
this module is described in term of the symmetric tensor square $Sym^2_R(M)$,
the degree 2 component $\Gamma^2_R(M)$ of the divided power algebra over $M$,
and the ideal $I_2$ of $R$ generated by the elements $r^2-r$, $r\in R$. The
latter is shown to represent quadratic derivations on $R$ which arise in the
theory of modules over square rings. This allows to extend the classical notion
of nilpotent $R$-group of class 2 with coefficients in a 2-binomial ring $R$ to
any ring $R$. We provide a functorial presentation of $I_2$ and several exact
sequences embedding the modules $P^2_R(M)$ and $\Gamma^2_R(M)$.
