Differential calculus over commutative algebras

In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are:
formula_14
where the bracket formula_15 is defined as the commutator
formula_16
Denoting the set of "k"th order linear differential operators from an formula_5-module formula_18 to an formula_5-module formula_20 with formula_21 we obtain a bi-functor with values in the category of formula_5-modules. Other natural concepts of calculus such as jet spaces, differential forms are then obtained as representing objects of the functors formula_23 and related functors.
Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects.
Replacing the real numbers formula_2 with any commutative ring, and the algebra formula_25 with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely used in algebraic geometry, differential geometry and secondary calculus. Moreover, the theory generalizes naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated concepts like the Berezin integral.