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

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

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...

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...

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

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...

Ring is a commutative ring. The action of a monoid (= commutative ring) R on an object (= ring) A of Ring is an R-algebra. The category of rings has a number...

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...

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...

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

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...

In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces...

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

is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the...

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...

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...

and prime ideals are both primary and semiprime. An ideal P of a commutative ring R is prime if it has the following two properties: If a and b are two...

entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be...

nonzero ring R in which every nonzero element is a unit (that is, R× = R ∖ {0}) is called a division ring (or a skew-field). A commutative division ring is...

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...

whose operation is commutative; a commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.) However, in the...

mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that...

In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R {\displaystyle R} is the set of all prime ideals of R {\displaystyle...

adjoint" to Hom. The tensor product of two modules A and B over a commutative ring R is defined in exactly the same way as the tensor product of vector...

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...

polynomial ring R[X] over a Noetherian ring R is Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again...

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...

left R-modules and Ab is the category of abelian groups. When the ring R is commutative, Ab is advantageously replaced by R-Mod in the preceding characterization...

-graded ring. If I is an ideal in a commutative ring R, then ⨁ n = 0 ∞ I n / I n + 1 {\textstyle \bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}} is a graded ring called...