In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or, more generally over a commutative ring) with an extra layer of structure, known as a gradation (or grading).
The grading is a direct sum decomposition of the algebra into modules indexed by a monoid, such that the product of two elements belonging to two summands of the grading results in an element in the summand indexed by the sum of the indices. The monoid is often the set of the non-negative integers using ordinary addition, but it can be any monoid. For example a finite group grades its own group algebra.
The term graded ring is sometimes used for the analogous grading of a ring. A graded ring could also be viewed as a graded Z-algebra.
The notion of a graded module is the generalization of graded vector spaces.
| Algebraic structures |
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Group-like structures
Semigroup and Monoid Quasigroup and Loop Abelian group |
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Ring-like structures
Semiring Near-ring Ring Commutative ring Integral domain Field |
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Lattice-like structures
Semilattice Lattice Map of lattices |
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Module-like structures
Group with operators Module Vector space |
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Algebra-like structures
Algebra Associative algebra Non-associative algebra Graded algebra Bialgebra |
Read more about Graded Algebra: Graded Ring, Graded Module, Graded Algebra, G-graded Rings and Algebras, Anticommutativity
Famous quotes containing the words graded and/or algebra:
“I don’t want to be graded on a curve.”
—Mary Carillo (b. 1957)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)