Editing Graded ring (section)

== ''G''-graded rings and algebras ==
The above definitions have been generalized to rings graded using any [[monoid]] ''G'' as an index set.  A '''''G''-graded ring''' ''R'' is a ring with a direct sum decomposition
: <math>R = \bigoplus_{i\in G}R_i </math>
such that
: <math> R_i R_j \subseteq R_{i \cdot j}. </math>
Elements of ''R'' that lie inside <math>R_i</math> for some <math>i \in G</math> are said to be '''homogeneous''' of '''grade''' ''i''.

The previously defined notion of "graded ring" now becomes the same thing as an <math>\N</math>-graded ring, where <math>\N</math> is the monoid of [[natural number]]s under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set <math>\N</math> with any monoid ''G''.

Remarks:
* If we do not require that the ring have an [[identity element]], [[semigroup]]s may replace monoids.

Examples:
* A [[group (mathematics)|group]] naturally grades the corresponding [[group ring]]; similarly, [[monoid ring]]s are graded by the corresponding monoid.
* An (associative) [[superalgebra]] is another term for a [[cyclic group|<math>\Z_2</math>]]-graded algebra. Examples include [[Clifford algebra]]s. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

=== Anticommutativity ===
Some graded rings (or algebras) are endowed with an [[anticommutative]] structure.  This notion requires a [[Monoid#Monoid homomorphisms|homomorphism]] of the monoid of the gradation into the additive monoid of <math>\Z/2\Z</math>, the field with two elements.  Specifically, a '''signed monoid''' consists of a pair <math>(\Gamma, \varepsilon)</math> where <math>\Gamma</math> is a monoid and <math>\varepsilon \colon \Gamma \to\Z/2\Z</math> is a homomorphism of additive monoids.  An '''anticommutative <math>\Gamma</math>-graded ring''' is a ring ''A'' graded with respect to <math>\Gamma</math> such that:
: <math>xy=(-1)^{\varepsilon (\deg x) \varepsilon (\deg y)}yx ,</math>
for all homogeneous elements ''x'' and ''y''.

=== Examples ===
* An [[exterior algebra]] is an example of an anticommutative algebra, graded with respect to the structure <math>(\Z, \varepsilon)</math> where <math>\varepsilon \colon \Z \to\Z/2\Z</math> is the quotient map.
* A [[supercommutative algebra]] (sometimes called a '''skew-commutative associative ring''') is the same thing as an anticommutative <math>(\Z, \varepsilon)</math>-graded algebra, where <math>\varepsilon</math> is the [[identity map]] of the additive structure of {{tmath|1= \Z/2\Z }}.