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

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...

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...

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...

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

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist...

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

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...

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...

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

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...

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

commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings,...

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...

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...

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....

is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers...

algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism that is also R-linear. The function f : Z/6Z → Z/6Z...

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...

An associative algebra A {\displaystyle A} over a commutative ring R {\displaystyle R} is a ring itself. If I {\displaystyle I} is an ideal in A {\displaystyle...

associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings. An algebra is unital or unitary if it has an identity element...

{\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...

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

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}...

differential graded algebras over Q {\displaystyle \mathbb {Q} } . Example: The Witten genus may be realized as a morphism of commutative ring spectra MString...

mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal group...

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality...

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"...

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...