Graded ring

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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.

Contents

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 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:

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 .

Basic examples

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 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.

Invariants of graded modules

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.

Graded algebra

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:

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.)

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 ringR is a ring with a direct sum decomposition

such that

Elements of R that lie inside for some are said to be homogeneous of gradei.

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:

Examples:

Anticommutativity

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.

Examples

Graded monoid

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.

Power series indexed by 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.

Example

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.

See also

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References

  1. 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.

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.