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In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a grading. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java...
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...
In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...
Graded algebra A graded algebra A is an algebra that has a direct sum decomposition In ring theory, an algebra over a base ring is a generalization of the concept of associative algebra. ...
In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
 such that  Elements of An are known as homogeneous elements of degree n. An ideal, or other set in A, is homogeneous if for every element a it contains, the homogeneous parts of a are also contained in it. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...
Since rings may be regarded as Z-algebras, a graded ring is defined to be a graded Z-algebra. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...
Examples of graded algebras are common in mathematics:
Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra and algebraic topology. One example is the close relationship between homogeneous polynomials and projective varieties. In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...
In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ...
In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ...
In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. ...
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...
In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
This article is about algebraic varieties. ...
G-graded algebra We can generalize the definition of a graded algebra to an arbitrary monoid G as an index set. A G-graded algebra A is an algebra with a direct sum decomposition In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
In ring theory, an algebra over a base ring is a generalization of the concept of associative algebra. ...
In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
 such that  A graded algebra is then the same thing as a N-graded algebra, where N is the monoid of natural numbers. In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...
(If we don't require that the ring has an identity element, we can extend the definition from monoids to semigroups. In mathematics, a semigroup is a set with an associative binary operation on it. ...
Examples of G-graded algebras include:
- The group ring of a group is naturally graded by that group; similarly, monoid rings are graded by the corresponding monoid.
- A superalgebra is another term for a Z2-graded algebra. Clifford algebras are a common family of examples. Here the homogeneous elements are either even (degree 0) or odd (degree 1).
Category theoretically, a G-graded algebra A is an object in the category of G-graded vector spaces together with a morphism  of the degree of the identity of G. In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described...
In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. ...
In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2). ...
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ...
Clifford algebras are a type of associative algebra in mathematics. ...
In mathematics, K-Z2Vect is the category with objects are Z2_graded vector spaces over the givenfield K and with morphisms the even and odd linear transformations between two Z2-graded vector spaces. ...
Graded modules The corresponding idea in module theory is that of a graded module, namely a module M over A such that also In abstract algebra, a module is a generalization of a vector space. ...
In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ...
 and  This idea is much used in commutative algebra, and elsewhere, to define under mild hypotheses a Hilbert function, namely the length of Mn as a function of n. Again under mild hypotheses of finiteness, this function is a polynomial, the Hilbert polynomial, for all large enough values of n (see also Hilbert-Samuel polynomial). In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ...
In abstract algebra, the length of a module is a measure of the modules size. It is defined as the length of the longest ascending chain of submodules and is a generalization of the concept of dimension for vector spaces. ...
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