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...

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

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...

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

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

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...

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...

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...

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

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...

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...

principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields An integral domain is a nonzero commutative ring in which the product of any two nonzero...

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

is commutative in every vector space and in every algebra. Union and intersection are commutative operations on sets. "And" and "or" are commutative logical...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials...

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...

of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space...

Noetherian ring is Noetherian. Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two...

problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme...

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

mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings...

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...

g(x). This makes these functions a F-commutative algebra. For having a field of functions, one must consider algebras of functions that are integral domains...

Bourbaki, N. (1998). Algebra I, Chapters 1–3. Springer. Eisenbud, David (1995). Commutative algebra with a view toward algebraic geometry. Graduate Texts...

A Banach algebra is called unital if it has an identity element for the multiplication whose norm is 1 , {\displaystyle 1,} and commutative if its multiplication...

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

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

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