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

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

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...

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...

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...

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

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...

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...

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

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...

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...

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

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

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...

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

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...

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...

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...

Introduction to Commutative Algebra. Perseus Books. ISBN 0-201-00361-9. Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.)....

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...

the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field...

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...

are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are...

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

and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves...