Graded ring#Graded algebra

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

: $R\; =\; \backslash bigoplus\_\{n=0\}^\backslash infty\; R\_n\; =\; R\_0\; \backslash oplus\; R\_1\; \backslash oplus\; R\_2\; \backslash oplus\; \backslash cdots$

of

additive groups, such that

: $R\_mR\_n\; \backslash subseteq\; R\_\{m+n\}$

for all nonnegative integers $m$ and {{tmath|1= n }}.

A nonzero element of $R\_n$ is said to be homogeneous of degree {{tmath|1= n }}. By definition of a direct sum, every nonzero element $a$ of $R$ can be uniquely written as a sum $a=a\_0+a\_1+\backslash cdots\; +a\_n$ where each $a\_i$ is either 0 or homogeneous of degree {{tmath|1= i }}. The nonzero $a\_i$ are the homogeneous components of {{tmath|1= a }}.

Some basic properties are:

• $R\_0$ is a subring of {{tmath|1= R }}; in particular, the multiplicative identity $1$ is a homogeneous element of degree zero.
• For any $n$, $R\_n$ is a two-sided {{tmath|1= R_0 }}-module, and the direct sum decomposition is a direct sum of {{tmath|1= R_0 }}-modules.
• $R$ is an associative algebra.

An ideal $I\backslash subseteq\; R$ is homogeneous, if for every {{tmath|1= a \in I }}, the homogeneous components of $a$ also belong to {{tmath|1= I }}. (Equivalently, if it is a graded submodule of {{tmath|1= R }}; see {{section link||Graded module}}.) The intersection of a homogeneous ideal $I$ with $R\_n$ is an {{tmath|1= R_0 }}-submodule of $R\_n$ called the homogeneous part of degree $n$ of {{tmath|1= I }}. A homogeneous ideal is the direct sum of its homogeneous parts.

If $I$ is a two-sided homogeneous ideal in {{tmath|1= R }}, then $R/I$ is also a graded ring, decomposed as

: $R/I\; =\; \backslash bigoplus\_\{n=0\}^\backslash infty\; R\_n/I\_n,$

where $I\_n$ is the homogeneous part of degree $n$ of {{tmath|1= I }}.

• Any (non-graded) ring R can be given a gradation by letting {{tmath|1= R_0=R }}, and $R\_i=0$ for i ≠ 0. This is called the trivial gradation on R.
• The polynomial ring $R\; =\; k[t\_1,\; \backslash ldots,\; t\_n]$ is graded by degree: it is a direct sum of $R\_i$ 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 $\backslash Z$-graded ring.
• If I is an ideal in a commutative ring R, then $\backslash bigoplus\_\{n=0\}^\{\backslash infty\}\; I^n/I^\{n+1\}$ 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^{{hairsp}}i(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 is $\backslash bigoplus\_\{i\; =\; 0\}^\backslash infty\; H^i(X;\; R)$ 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

: $M\; =\; \backslash bigoplus\_\{i\backslash in\; \backslash mathbb\{N\}\}M\_i\; ,$

and

: $R\_iM\_j\; \backslash subseteq\; M\_\{i+j\}$

for every {{mvar|i}} and {{mvar|j}}.

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, the direct sum $\backslash bigoplus\_\{n=0\}^\{\backslash infty\}\; I^n\; M/I^\{n+1\}\; M$ is a graded module over the associated graded ring $\backslash bigoplus\_0^\{\backslash infty\}\; I^n/I^\{n+1\}$.

A morphism $f:\; N\; \backslash to\; M$ of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the underlying modules that respects grading; i.e., {{tmath|1= f(N_i) \subseteq M_i }}. 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 {{tmath|1= N_i = N \cap M_i }}. 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 $M$, the $\backslash ell$-twist of $M$ is a graded module defined by $M(\backslash ell)\_n\; =\; M\_\{n+\backslash ell\}$ (cf. Serre's twisting sheaf in algebraic geometry).

Let M and N be graded modules. If $f\backslash colon\; M\; \backslash to\; N$ is a morphism of modules, then f is said to have degree d if $f(M\_n)\; \backslash subseteq\; N\_\{n+d\}$. 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 {{tmath|1= P(M, t) \in \Z[\![t]\!] }}:

: $P(M,\; t)\; =\; \backslash sum\; \backslash ell(M\_n)\; t^n$

(assuming $\backslash ell(M\_n)$ 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 {{tmath|1= k[x_0, \dots, x_n] }}, k a field, and M a finitely generated graded module over it. Then the function $n\; \backslash mapsto\; \backslash dim\_k\; M\_n$ is called the Hilbert function of M. The function coincides with the integer-valued polynomial for large n called the Hilbert polynomial of M.

{{seealso|Graded Lie 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 is of degree 0). Thus, $R\backslash subseteq\; A\_0$ and the graded pieces $A\_i$ are R-modules.

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

: $R\_iA\_j\; \backslash subseteq\; A\_\{i+j\}$

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 $T^\{\backslash bullet\}\; V$ of a vector space V. The homogeneous elements of degree n are the tensors of order n, {{tmath|1= T^{n} V }}.
• The exterior algebra $\backslash textstyle\backslash bigwedge\backslash nolimits^\{\backslash bullet\}\; V$ and the symmetric algebra $S^\{\backslash bullet\}\; V$ are also graded algebras.
• The cohomology ring $H^\{\backslash bullet\}$ in any cohomology theory is also graded, being the direct sum of the cohomology groups {{tmath|1= H^n }}.

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

: $R\; =\; \backslash bigoplus\_\{i\backslash in\; G\}R\_i$

such that

: $R\_i\; R\_j\; \backslash subseteq\; R\_\{i\; \backslash cdot\; j\}.$

Elements of R that lie inside $R\_i$ for some $i\; \backslash in\; G$ are said to be homogeneous of grade i.

The previously defined notion of "graded ring" now becomes the same thing as an $\backslash N$-graded ring, where $\backslash N$ is the monoid of natural numbers under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set $\backslash N$ 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 $\backslash Z\_2$-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 $\backslash Z/2\backslash Z$, the field with two elements. Specifically, a signed monoid consists of a pair $(\backslash Gamma,\; \backslash varepsilon)$ where $\backslash Gamma$ is a monoid and $\backslash varepsilon\; \backslash colon\; \backslash Gamma\; \backslash to\backslash Z/2\backslash Z$ is a homomorphism of additive monoids. An anticommutative $\backslash Gamma$-graded ring is a ring A graded with respect to $\backslash Gamma$ such that:

: $xy=(-1)^\{\backslash varepsilon\; (\backslash deg\; x)\; \backslash varepsilon\; (\backslash deg\; y)\}yx\; ,$

for all homogeneous elements x and y.

• An exterior algebra is an example of an anticommutative algebra, graded with respect to the structure $(\backslash Z,\; \backslash varepsilon)$ where $\backslash varepsilon\; \backslash colon\; \backslash Z\; \backslash to\backslash Z/2\backslash Z$ is the quotient map.
• A supercommutative algebra (sometimes called a skew-commutative associative ring) is the same thing as an anticommutative $(\backslash Z,\; \backslash varepsilon)$-graded algebra, where $\backslash varepsilon$ is the identity map of the additive structure of {{tmath|1= \Z/2\Z }}.

Intuitively, a graded monoid is the subset of a graded ring, $\backslash bigoplus\_\{n\backslash in\; \backslash mathbb\; N\_0\}R\_n$, generated by the $R\_n$'s, without using the additive part. That is, the set of elements of the graded monoid is $\backslash bigcup\_\{n\backslash in\backslash mathbb\; N\_0\}R\_n$.

Formally, a graded monoid{{cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | translator-first=Reuben|translator-last=Thomas | publisher=Cambridge University Press | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 | chapter = Part II: The power of algebra | page=384 }} is a monoid $(M,\backslash cdot)$, with a gradation function $\backslash phi:M\backslash to\backslash mathbb\; N\_0$ such that $\backslash phi(m\backslash cdot\; m\text{'})=\backslash phi(m)+\backslash phi(m\text{'})$. Note that the gradation of $1\_M$ is necessarily 0. Some authors request furthermore that $\backslash phi(m)\backslash ne\; 0$

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 $g^n$ 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 $n+1$ (for $g=1$) or $\backslash frac\{g^\{n+1\}-1\}\{g-1\}$ else. Indeed, each such element is the product of at most n elements of G, and only $\backslash frac\{g^\{n+1\}-1\}\{g-1\}$ 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.

{{see also|Novikov ring}}

These notions allow us to extend the notion of power series ring. Instead of the indexing family being $\backslash mathbb\; N$, 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 $(K,+\_K,\backslash times\_K)$ be an arbitrary semiring and $(R,\backslash cdot,\backslash phi)$ a graded monoid. Then $K\backslash langle\backslash langle\; R\backslash rangle\backslash rangle$ 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 $s,s\text{'}\backslash in\; K\backslash langle\backslash langle\; R\backslash rangle\backslash rangle$ is defined pointwise, it is the function sending $m\backslash in\; R$ to $s(m)+\_Ks\text{'}(m)$, and the product is the function sending $m\backslash in\; R$ to the infinite sum $\backslash sum\_\{p,q\; \backslash in\; R\; \backslash atop\; p\; \backslash cdot\; q=m\}s(p)\backslash times\_K\; s\text{'}(q)$. This sum is correctly defined (i.e., finite) because, for each m, there are only a finite number of pairs {{nowrap|(p, q)}} such that {{nowrap|1=pq = m}}.

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.

• Associated graded ring
• Differential graded algebra
• Filtered algebra, a generalization
• Graded (mathematics)
• Graded category
• Graded vector space
• Tensor algebra
• Differential graded module

{{reflist}}

{{refbegin}}

• {{Lang Algebra}}.
• {{cite book |author-link=Nicolas Bourbaki |first=N. |last=Bourbaki |chapter=Ch. 1–3, 3 §3 |title=Algebra I |publisher= Springer|location= |year=1974 |isbn=978-3-540-64243-5 }}
• {{cite journal |first=J. |last=Steenbrink |title=Intersection form for quasi-homogeneous singularities |journal=Compositio Mathematica |volume=34 |issue=2 |pages=211–223 See p. 211 |year=1977 |issn=0010-437X |url=http://archive.numdam.org/article/CM_1977__34_2_211_0.pdf}}
• {{cite book |first=H. |last=Matsumura |title=Commutative Ring Theory |url=https://books.google.com/books?id=J68-BAAAQBAJ |date=1989 |publisher=Cambridge University Press |isbn=978-1-107-71712-1 |translator-first=M. |translator-last=Reid |edition=2nd |series=Cambridge Studies in Advanced Mathematics |volume=8 |chapter=5 Dimension theory §S3 Graded rings, the Hilbert function and the Samuel function}}

{{refend}}

Category:Algebras

Category:Ring theory