stylistic:style_transfer
Graded-commutative rings

A graded ring R = ⨁ R is called graded-commutative if, for all homogeneous elements a and b,

If the R are connected by differentials ∂ such that an abstract form of the product rule holds, i.e.,

R is called a commutative differential graded algebra (cdga). An example is the complex of differential forms on a manifold, with the multiplication given by the exterior product, is a cdga. The cohomology of a cdga is a graded-commutative ring, sometimes referred to as the cohomology ring. A broad range examples of graded rings arises in this way. For example, the Lazard ring is the ring of cobordism classes of complex manifolds.

A graded-commutative ring with respect to a grading by Z /2 (as opposed to Z) is called a superalgebra.

A related notion is an almost commutative ring, which means that R is filtered in such a way that the associated graded ring

is commutative. An example is the Weyl algebra and more general rings of differential operators.