Graded Algebra - G-graded Rings and Algebras

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).