Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations.

In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

* Any commutative ring R is an algebra over itself, or any subring of R.

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order ( the axiom of commutativity ).

Many binary operations of interest in both algebra and formal logic are commutative or associative.

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

* Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.

* Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional.

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

As an algebra, a unital commutative Banach algebra is semisimple ( i. e., its Jacobson radical is zero ) if and only if its Gelfand representation has trivial kernel.

An important example of such an algebra is a commutative C *- algebra.

In fact, when A is a commutative unital C *- algebra, the Gelfand representation is then an isometric *- isomorphism between A and C ( Δ ( A )).

* The structure space of a commutative unital Banach algebra is a compact Hausdorff space.

In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

In abstract algebra, an integral domain is a commutative ring that has no zero divisors, and which is not the trivial ring

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

The ring theory, which was firmly established during the 1920s by Emmy Noether and Wolfgang Krull, acquires a distinctly different flavor depending whether it allows rings to be commutative or not.

If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module.

The proof of the general case requires some commutative algebra, namely the fact, that the Krull dimension of k is one — see Krull's principal ideal theorem ).

In the 1930s, Wolfgang Krull turned things around and took a radical step: start with any commutative ring, consider the set of its prime ideals, turn it into a topological space by introducing the Zariski topology, and study the algebraic geometry of these quite general objects.

They concern a number of interrelated ( sometimes surprisingly so ) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth.

A local Cohen – Macaulay ring is defined as a commutative noetherian local ring with Krull dimension equal to its depth.

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.

For a Noetherian commutative local ring of Krull dimension, the following are equivalent:

In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull ( 1899 – 1971 ), gives a bound on the height of a principal ideal in a Noetherian ring.

In the theory of modules over a commutative ring R, when R has Krull dimension ≥ 2, it can be useful to treat modules M and N as pseudo-isomorphic if M / N has support of codimension at least two.

Wolfgang Krull ( 26 August 1899-12 April 1971 ) was a German mathematician working in the field of commutative algebra.

In mathematics, more specifically modern algebra and commutative algebra, Nakayama's lemma also known as the Krull – Azumaya theorem governs the interaction between the Jacobson radical of a ring ( typically a commutative ring ) and its finitely generated modules.

The lemma is named after the Japanese mathematician Tadashi Nakayama and introduced in its present form in, although it was first discovered in the special case of ideals in a commutative ring by Wolfgang Krull and then in general by Goro Azumaya ( 1951 ).

Serre proved that a commutative Noetherian local ring A is regular if and only if it has finite global dimension, in which case the global dimension coincides with the Krull dimension of A.

If R is a field, the cycle space is a vector space over R with dimension m-n + c, where c is the number of connected components of G. If R is any commutative ring, the cycle space is a free R-module with rank m-n + c.

For example, for ( 1 + 1 )- dimensional bordisms ( 2-dimensional bordisms between 1-dimensional manifolds ), the map associated with a pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit ( trace ) or unit ( scalars ), depending on grouping of boundary, and thus ( 1 + 1 )- dimension TQFTs correspond to Frobenius algebras.

This discussion ( aside from statements referring to dimension and Lie group ) applies if we replace R by any commutative ring A.

In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module.

A ( not necessarily commutative ) ring R is called Gorenstein if R has finite injective dimension both as a left R-module and as a right R-module.

In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group.

Note that the two-dimensional case is the only non-trivial ( e. g. one dimension ) case where the rotation matrices group is commutative, so that it does not matter the order in which multiple rotations are performed.

Moreover if we define scalar multiplication in the obvious manner, the split-complex numbers actually form a commutative and associative algebra over the reals of dimension two.

the tensor product of End ( A ) with the rational number field Q, should contain a commutative subring of dimension 2d over Z.

By a theorem of Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local rings those rings which are regular.

In particular, a commutative principal ideal domain which is not a field has global dimension one.

* when A is a commutative Noetherian local ring with maximal ideal m, the projective dimension of the residue field A / m.