G-graded Rings and Algebras
Above definitions have been generalized to gradings ring using any monoid G as an index set. A G-graded ring A is a ring with a direct sum decomposition
such that
The notion of "graded ring" now becomes the same thing as a N-graded ring, where N is the monoid of non-negative integers under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set N with any monoid G.
Remarks:
- If we do not require that the ring have an identity element, semigroups may replace monoids.
Examples:
- A group naturally grades the corresponding group ring; similarly, monoid rings are graded by the corresponding monoid.
- A superalgebra is another term for a Z2-graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).
Read more about this topic: Graded Algebra
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