Editing Commutative ring (section)

== Generalizations ==

=== Graded-commutative rings ===
[[File:Pair_of_pants.png|thumb|A [[pair of pants (mathematics)|pair of pants]] is a [[cobordism]] between a circle and two disjoint circles. Cobordism classes, with the [[cartesian product]] as multiplication and [[disjoint union]] as the sum, form the [[cobordism ring]].]]
A [[graded ring]] {{nowrap|1=''R'' = ⨁<sub>''i''∊'''Z'''</sub> ''R''<sub>''i''</sub>}} is called [[graded-commutative ring|graded-commutative]] if, for all homogeneous elements ''a'' and ''b'',
{{block indent|1= ''ab'' = (&minus;1)<sup>deg ''a'' ⋅ deg ''b''</sup> ''ba''. }}
If the ''R''<sub>''i''</sub> are connected by differentials ∂ such that an abstract form of the [[product rule]] holds, i.e.,
{{block indent|1= ∂(''ab'') = ∂(''a'')''b'' + (&minus;1)<sup>deg ''a''</sup>a∂(''b''), }}
''R'' is called a [[differential graded algebra|commutative differential graded algebra]] (cdga). An example is the complex of [[differential form]]s on a [[manifold (mathematics)|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's universal ring|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 [[filtration (mathematics)|filtered]] in such a way that the associated graded ring
{{block indent|1= gr ''R'' := ⨁ ''F''<sub>''i''</sub>''R'' / ⨁ ''F''<sub>''i''&minus;1</sub>''R'' }}
is commutative. An example is the [[Weyl algebra]] and more general rings of [[differential operator]]s.

=== Simplicial commutative rings ===
A [[simplicial commutative ring]] is a [[simplicial object]] in the category of commutative rings. They are building blocks for (connective) [[derived algebraic geometry]]. A closely related but more general notion is that of [[E-infinity ring|E<sub>∞</sub>-ring]].