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

In 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...

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

some subjects such as algebraic geometry, unital associative commutative algebra. Replacing the field of scalars by a commutative ring leads to the more...

involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number...

necessarily commutative" for noncommutative rings. An algebra is unital or unitary if it has an identity element e with ex = x = xe for all x in the algebra. For...

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...

Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties...

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry...

numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures...

grading, graded-commutative or, if the supercommutativity is understood, simply commutative. Any commutative algebra is a supercommutative algebra if given the...

ring is commutative (that is, its multiplication is a commutative operation) has profound implications on its properties. Commutative algebra, the theory...

mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and...

multiplication is commutative. Any Banach algebra A {\displaystyle A} (whether it is unital or not) can be embedded isometrically into a unital Banach algebra A e {\displaystyle...

Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of...

glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary...

ring may be regarded as a free commutative algebra. For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X1,...,Xn}...

fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique...

geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from...

homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes...

over a commutative ring. The collection of all structures of a given type (same operations and same laws) is called a variety in universal algebra; this...

result. It is said that commutative diagrams play the role in category theory that equations play in algebra. A commutative diagram often consists of...

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...

by using the continuous functional calculus or by reduction to commutative C*-algebras. In the latter case, we can use the fact that the structure of...

and n. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element x {\displaystyle x} ,...

all commutative rings. This category is one of the central objects of study in the subject of commutative algebra. Any ring can be made commutative by...

mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A {\displaystyle A} over a field...

generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which x y {\displaystyle...

} As for algebras, one can replace the underlying field K with a commutative ring R in the above definition. The definition of Hopf algebra is self-dual...

algebras are non-commutative rings. An operator algebra is typically required to be closed in a specified operator topology inside the whole algebra of...