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.
A graded module that is also a gradedring is called a graded algebra. A gradedring could also be viewed as a graded Z {\displaystyle \mathbb {Z} } -algebra...
In mathematics, the associated gradedring of a ring R with respect to a proper ideal I is the gradedring: gr I R = ⨁ n = 0 ∞ I n / I n + 1 {\displaystyle...
cohomology of a cdga is a graded-commutative ring, sometimes referred to as the cohomology ring. A broad range examples of gradedrings arises in this way....
polynomials, gradedrings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of polynomial...
into a ring. In fact, it is naturally an N-gradedring with the nonnegative integer k serving as the degree. The cup product respects this grading. The...
mathematics, the pluricanonical ring of an algebraic variety V (which is nonsingular), or of a complex manifold, is the gradedring R ( V , K ) = R ( V , K V...
distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a gradedring, H∗(X), called the cohomology ring. The...
B]]]\right)+\cdots \right).} When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components...
performance Grade, the number of the year a student has reached in a given educational stage Grade (slope), the steepness of a slope Graded voting Grade or grading...
A corona ring, more correctly referred to as an anti-corona ring, is a toroid of conductive material, usually metal, which is attached to a terminal or...
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite...
{\displaystyle {\mathfrak {m}}} -primary ideals. The dimension of the gradedring gr m ( R ) = ⨁ k ≥ 0 m k / m k + 1 {\displaystyle \textstyle \operatorname...
_{*}A} is a gradedring over π 0 A {\displaystyle \pi _{0}A} .) A topology-counterpart of this notion is a commutative ring spectrum. The ring of polynomial...
context of gradedrings R, the canonical module of a Gorenstein ring R is isomorphic to R with some degree shift. For a Gorenstein local ring (R, m, k)...
In mathematics, the irrelevant ideal is the ideal of a gradedring generated by the homogeneous elements of degree greater than zero. It corresponds to...
In mathematics, the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is the gradedring generated by the modular forms...
{\displaystyle H^{*}(X,R)=\bigoplus _{i}H^{i}(X,R)} into a gradedring, called the cohomology ring of X. It is graded-commutative in the sense that: u v = ( − 1 ) i...
{g}}} is a both left and right Noetherian ring; this follows from the fact that the associated gradedring of U is a quotient of Sym ( g ) {\displaystyle...
mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is...
i > 1 in order to be symmetric. Unlike the whole power series ring, the subring ΛR is graded by the total degree of monomials: due to condition 2, every...
In algebra, a filtered ring A is said to be almost commutative if the associated gradedring gr A = ⊕ A i / A i − 1 {\displaystyle \operatorname {gr}...
ring of V. Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over...
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