Quadratic maps between modules

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 . The structure of this module is described in term of the symmetric tensor square , the degree 2 component of the divided power algebra over M, and the ideal I2 of R generated by the elements r2−r, r∈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 I2 and several exact sequences embedding the modules and .