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In mathematics, in particular abstract algebra, a **graded ring** is a ring such that the underlying additive group is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as **gradation** or **grading**.

- First properties
- Basic examples
- Graded module
- Invariants of graded modules
- Graded algebra
- G-graded rings and algebras
- Anticommutativity
- Examples
- Graded monoid
- Power series indexed by a graded monoid
- Example
- See also
- References

A **graded module** is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a **graded algebra**. A graded ring could also be viewed as a graded -algebra.

The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra.

Generally, the index set of a graded ring is supposed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article.

A graded ring is a ring that is decomposed into a direct sum

of additive groups, such that

for all nonnegative integers and .

A nonzero element of is said to be *homogeneous* of *degree*. By definition of a direct sum, every nonzero element of can be uniquely written as a sum where each is either 0 or homogeneous of degree . The nonzero are the *homogeneous components* of .

Some basic properties are:

- is a subring of ; in particular, the multiplicative identity is an homogeneous element of degree zero.
- For any , is a two-sided -module, and the direct sum decomposition is a direct sum of -modules.
- is an associative -algebra.

An ideal is *homogeneous*, if for every , the homogeneous components of also belong to (Equivalently, if it is a graded submodule of ; see § Graded module.) The intersection of a homogeneous ideal with is an -submodule of called the *homogeneous part* of degree of . A homogeneous ideal is the direct sum of its homogeneous parts.

If is a two-sided homogeneous ideal in , then is also a graded ring, decomposed as

where is the homogeneous part of degree of .

- Any (non-graded) ring
*R*can be given a gradation by letting , and for*i*≠ 0. This is called the**trivial gradation**on*R*. - The polynomial ring is graded by degree: it is a direct sum of consisting of homogeneous polynomials of degree
*i*. - Let
*S*be the set of all nonzero homogeneous elements in a graded integral domain*R*. Then the localization of*R*with respect to*S*is a -graded ring. - If
*I*is an ideal in a commutative ring*R*, then is a graded ring called the associated graded ring of*R*along*I*; geometrically, it is the coordinate ring of the normal cone along the subvariety defined by*I*. - Let
*X*be a topological space,*H*the^{i}(X; R)*i*th cohomology group with coefficients in a ring*R*. Then*H*, the cohomology ring of^{*}(X; R)*X*with coefficients in*R*, is a graded ring whose underlying group is with the multiplicative structure given by the cup product.

The corresponding idea in module theory is that of a **graded module**, namely a left module *M* over a graded ring *R* such that also

and

**Example**: a graded vector space is an example of a graded module over a field (with the field having trivial grading).

**Example**: a graded ring is a graded module over itself. An ideal in a graded ring is homogeneous if and only if it is a graded submodule. The annihilator of a graded module is a homogeneous ideal.

**Example**: Given an ideal *I* in a commutative ring *R* and an *R*-module *M*, the direct sum is a graded module over the associated graded ring .

A morphism between graded modules, called a **graded morphism**, is a morphism of underlying modules that respects grading; i.e., . A **graded submodule** is a submodule that is a graded module in own right and such that the set-theoretic inclusion is a morphism of graded modules. Explicitly, a graded module *N* is a graded submodule of *M* if and only if it is a submodule of *M* and satisfies . The kernel and the image of a morphism of graded modules are graded submodules.

Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the center is the same as to give the structure of a graded algebra to the latter ring.

Given a graded module , the -twist of is a graded module defined by . (cf. Serre's twisting sheaf in algebraic geometry.)

Let *M* and *N* be graded modules. If is a morphism of modules, then *f* is said to have degree *d* if . An exterior derivative of differential forms in differential geometry is an example of such a morphism having degree 1.

Given a graded module *M* over a commutative graded ring *R*, one can associate the formal power series :

(assuming are finite.) It is called the Hilbert–Poincaré series of *M*.

A graded module is said to be finitely generated if the underlying module is finitely generated. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)

Suppose *R* is a polynomial ring , *k* a field, and *M* a finitely generated graded module over it. Then the function is called the Hilbert function of *M*. The function coincides with the integer-valued polynomial for large *n* called the Hilbert polynomial of *M*.

An algebra *A* over a ring *R* is a **graded algebra** if it is graded as a ring.

In the usual case where the ring *R* is not graded (in particular if *R* is a field), it is given the trivial grading (every element of *R* is of degree 0). Thus, and the graded pieces are *R*-modules.

In the case where the ring *R* is also a graded ring, then one requires that

In other words, we require *A* to be a graded left module over *R*.

Examples of graded algebras are common in mathematics:

- Polynomial rings. The homogeneous elements of degree
*n*are exactly the homogeneous polynomials of degree*n*. - The tensor algebra of a vector space
*V*. The homogeneous elements of degree*n*are the tensors of order*n*, . - The exterior algebra and the symmetric algebra are also graded algebras.
- The cohomology ring in any cohomology theory is also graded, being the direct sum of the cohomology groups .

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 (cf. homogeneous coordinate ring.)

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

such that

Elements of *R* that lie inside for some are said to be **homogeneous** of **grade***i*.

The previously defined notion of "graded ring" now becomes the same thing as an -graded ring, where 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 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.
- An (associative) superalgebra is another term for a -graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

Some graded rings (or algebras) are endowed with an anticommutative structure. This notion requires a homomorphism of the monoid of the gradation into the additive monoid of , the field with two elements. Specifically, a **signed monoid** consists of a pair where is a monoid and is a homomorphism of additive monoids. An **anticommutative -graded ring** is a ring *A* graded with respect to Γ such that:

for all homogeneous elements *x* and *y*.

- An exterior algebra is an example of an anticommutative algebra, graded with respect to the structure where is the quotient map.
- A supercommutative algebra (sometimes called a
**skew-commutative associative ring**) is the same thing as an anticommutative -graded algebra, where is the identity endomorphism of the additive structure of .

Intuitively, a graded monoid is the subset of a graded ring, , generated by the 's, without using the additive part. That is, the set of elements of the graded monoid is .

Formally, a graded monoid^{ [1] } is a monoid , with a gradation function such that . Note that the gradation of is necessarily 0. Some authors request furthermore that when **m** is not the identity.

Assuming the gradations of non-identity elements are non zero, the number of elements of gradation *n* is at most where *g* is the cardinality of a generating set *G* of the monoid. Therefore the number of elements of gradation *n* or less is at most (for ) or else. Indeed, each such element is the product of at most *n* elements of *G*, and only such products exist. Similarly, the identity element can not be written as the product of two non-identity elements. That is, there is no unit divisor in such a graded monoid.

This notions allows to extends the notion of power series ring. Instead of having the indexing family being , the indexing family could be any graded monoid, assuming that the number of elements of degree *n* is finite, for each integer *n*.

More formally, let be an arbitrary semiring and a graded monoid. Then denotes the semiring of power series with coefficients in *K* indexed by *R*. Its elements are functions from *R* to *K*. The sum of two elements is defined point-wise, it is the function sending to . And the product is the function sending to the infinite sum . This sum is correctly defined (i.e., finite) because, for each *m*, only a finite number of pairs (*p*, *q*) such that *pq* = *m* exist.

In formal language theory, given an alphabet *A*, the free monoid of words over *A* can be considered as a graded monoid, where the gradation of a word is its length.

In mathematics, specifically abstract algebra, the **isomorphism theorems** are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

In abstract algebra, the **direct sum** is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.

In ring theory, a branch of abstract algebra, a **commutative ring** is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

In mathematics, a **finitely generated module** is a module that has a finite generating set. A finitely generated module over a ring *R* may also be called a **finite R-module**,

In mathematics, the **annihilator** of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of S.

In mathematics, especially in the field of algebra, a **polynomial ring** or **polynomial algebra** is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics and theoretical physics, a **superalgebra** is a **Z**_{2}-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.

In mathematics, especially in the area of abstract algebra known as module theory, an **injective module** is a module *Q* that shares certain desirable properties with the **Z**-module **Q** of all rational numbers. Specifically, if *Q* is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module *Y*, then any module homomorphism from this submodule to *Q* can be extended to a homomorphism from all of *Y* to *Q*. This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in some detail in the textbook.

In mathematics, the **symmetric algebra***S*(*V*) on a vector space *V* over a field *K* is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. Here, "minimal" means that *S*(*V*) satisfies the following universal property: for every linear map f from V to a commutative algebra A, there is a unique algebra homomorphism *g* : *S*(*V*) → *A* such that *f* = *g* ∘ *i*, where i is the inclusion map of V in *S*(*V*).

In mathematics, a **graded vector space** is a vector space that has the extra structure of a *grading* or a *gradation*, which is a decomposition of the vector space into a direct sum of vector subspaces.

In mathematics, the **Lasker–Noether theorem** states that every Noetherian ring is a **Lasker ring**, which means that every ideal can be decomposed as an intersection, called **primary decomposition**, of finitely many *primary ideals*. The theorem was first proven by Emanuel Lasker (1905) for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether (1921).

In algebraic geometry, **Proj** is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental tool in scheme theory.

In mathematics, in particular abstract algebra and topology, a **differential graded algebra** is a graded algebra with an added chain complex structure that respects the algebra structure.

In mathematics, a **filtered algebra** is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.

In mathematics, a **graded Lie algebra** is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra.

In mathematics, the **Artin–Rees lemma** is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work.

In mathematics, in the field of abstract algebra, the **structure theorem for finitely generated modules over a principal ideal domain** is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

In mathematics, and in particular in the field of algebra, a **Hilbert–Poincaré series**, named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures. It is a formal power series in one indeterminate, say , where the coefficient of gives the dimension of the sub-structure of elements homogeneous of degree . It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a rational function of its argument .

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.

In mathematics, the term “**graded**” has a number of meanings, mostly related:

- ↑ Sakarovitch, Jacques (2009). "Part II: The power of algebra".
*Elements of automata theory*. Translated by Thomas, Reuben. Cambridge University Press. p. 384. ISBN 978-0-521-84425-3. Zbl 1188.68177.

- Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics,**211**(Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 . - Bourbaki, N. (1974). "Ch. 1–3, 3 §3".
*Algebra I*. ISBN 978-3-540-64243-5. - Steenbrink, J. (1977). "Intersection form for quasi-homogeneous singularities" (PDF).
*Compositio Mathematica*.**34**(2): 211–223 See p. 211. ISSN 0010-437X.

Matsumura, H. (1989). "5 Dimension theory §S3 Graded rings, the Hilbert function and the Samuel function". *Commutative Ring Theory*. Cambridge Studies in Advanced Mathematics. **8**. Translated by Reid, M. (2nd ed.). Cambridge University Press. ISBN 978-1-107-71712-1.

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