Graded ring on arvau

Wikipedia / Graded ring

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.

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.

First properties

Generally, the index set of a graded ring is assumed 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

and .

is said to be homogeneous of degree is either 0 or homogeneous of degree are the homogeneous components of.

Some basic properties are:

is a subring of is a homogeneous element of degree zero.
is a two-sided -module, and the direct sum decomposition is a direct sum of -modules.
is an associative R_0-algebra.

Ideals

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

Quotients

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

of .

Basic examples

Any (non-graded) ring R can be given a gradation by letting for i ≠ 0. This is called the trivial gradation onR.
The polynomial ring is graded by degree 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-graded ring.
If I is an ideal in a commutative ring R 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, Hi(X; R) the ith cohomology group with coefficients in a ring R. Then H*(X; R), the cohomology ring of X with coefficients in R, is a graded ring whose underlying group with the multiplicative structure given by the cup product.

Graded module

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

and

for every and .

Examples:

A graded vector space is an example of a graded module over a field (with the field having trivial grading).
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.
Given an ideal I in a commutative ring R and an R-module M.

A morphism of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the 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.

(cf. Serre's twisting sheaf in algebraic geometry).

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

Invariants of graded modules

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

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 is called the Hilbert function of M. The function coincides with the integer-valued polynomial for large n called the Hilbert polynomial of M.

Graded algebra

An associative 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 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).

Another example of a graded algebra which is non-commutative is the Leavitt path algebra-grading.

G-graded rings and algebras

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 are said to be homogeneous of grade i.

is the monoid of natural numbers 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 \Z_2-graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

Anticommutativity

Some graded rings (or algebras) are endowed with an anticommutative structure. This notion requires a homomorphism, the field with two elements. Specifically, a signed monoid is a homomorphism of additive monoids. An -graded ring is a ring A such that:

for all homogeneous elements x and y.

Examples

An exterior algebra is the quotient map.
A supercommutative algebra (sometimes called a skew-commutative associative ring is the identity map of the additive structure of .

Graded monoid

Intuitively, a graded monoid.

Formally, a graded monoid when m is not the identity.

Assuming the gradations of non-identity elements are non-zero, the number of elements of gradation n where g is the cardinality of a generating set G of the monoid. Therefore, the number of elements of gradation n else. Indeed, each such element is the product of at most n elements of G 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.

Power series indexed by a graded monoid

These notions allow us to extend the notion of power series ring, the indexing family could be any graded monoid, assuming that the number of elements of degree n is finite, for each integer n.

be an arbitrary semiring denotes the semiring of power series with coefficients in K indexed by R. Its elements are functions from R to K. This sum is correctly defined (i.e., finite) because, for each m, there are only a finite number of pairs such that .

Free monoid

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.