A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded ⁠ Z {\displaystyle \mathbb {Z} }...

In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous...

mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra...

mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different...

studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g....

examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major...

theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even"...

referred to as skew-commutative associative algebras to emphasize the anti-commutation, or, to emphasize the grading, graded-commutative or, if the supercommutativity...

underlying rings are graded-commutative rings, the tensor product A ⊗ R B {\displaystyle A\otimes _{R}B} becomes a graded commutative ring by defining...

Lie algebra Lie bracket of a Lie ring Subtraction Commutativity Commutator Exterior algebra Graded-commutative ring Operation (mathematics) Symmetry in...

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both...

multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers...

In algebra, a filtered ring A is said to be almost commutative if the associated graded ring gr ⁡ A = ⊕ A i / A i − 1 {\displaystyle \operatorname {gr}...

{\displaystyle \pi _{*}A=\oplus _{i\geq 0}\pi _{i}A} the structure of a graded-commutative graded ring as follows. By the Dold–Kan correspondence, π ∗ A {\displaystyle...

a b–1 ≠ b–1 a. A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite...

associative commutative algebra. Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Algebras...

right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics...

mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A...

beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether. Given a ring R, a left ideal...

space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian...

{\displaystyle A} are called graded functions on a graded manifold ( Z , A ) {\displaystyle (Z,A)} . They make up a graded commutative C ∞ ( Z ) {\displaystyle...

In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many...

In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring: gr I ⁡ R = ⨁ n = 0 ∞ I n / I n + 1 {\displaystyle...

R)=\bigoplus _{i}H^{i}(X,R)} into a graded ring, called the cohomology ring of X {\displaystyle X} . It is graded-commutative in the sense that: u v = ( − 1...

over a commutative ring R is a ring homomorphism that is also R-linear. The function f : Z/6Z → Z/6Z defined by f([a]6) = [4a]6 is not a ring homorphism...

does not hold in general for non-commutative Noetherian rings. Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not...

classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutative ring spectra...

zero ring is commutative. The element 0 in the zero ring is a unit, serving as its own multiplicative inverse. The unit group of the zero ring is the...

definition, any ring and any semifield is also a semiring. The non-negative elements of a commutative, discretely ordered ring form a commutative, discretely...

category of augmented differential graded commutative algebras. A graded Lie algebra is a Lie algebra that is graded as a vector space by a gradation compatible...