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.

Let R be a fixed commutative ring.

If A itself is commutative ( as a ring ) then it is called a commutative R-algebra.

* Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication.

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

( valid for any elements x, y of a commutative ring ),

* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

A ring homomorphism of commutative rings determines a morphism of Kähler differentials which sends an element dr to d ( f ( r )), the exterior differential of f ( r ).

Two ideals A and B in the commutative ring R are called coprime ( or comaximal ) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals ( a ) and ( b ) in the ring of integers Z are coprime if and only if a and b are coprime.

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

Provided the underlying scalars form a field ( more generally, a commutative ring with unity ), the definition below shows that such a function exists, and it can be shown to be unique.

The center of a division ring is commutative and therefore a field.

The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism of commutative rings.

The subcategory of commutative R-algebras can be characterized as the coslice category R / CRing where CRing is the category of commutative rings.

The entries can be numbers or expressions ( as happens when the determinant is used to define a characteristic polynomial ); the definition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner.

The latter case with the function f can be expressed by a commutative triangle.

It can be shown that in every such ring, multiplication is commutative, and every element is its own additive inverse.

* The ring of formal power series over a commutative ring R can be thought of as the inverse limit of the rings, indexed by the natural numbers as usually ordered, with the morphisms from to given by the natural projection.

The commutative law of addition can be used to rearrange terms into any preferred order.

If R is commutative, then one can associate to every polynomial P in R, a polynomial function f with domain and range equal to R ( more generally one can take domain and range to be the same unital associative algebra over R ).

A multiplicative identity is not required for the role of the integral domain ; this construction can be applied to any non-trivial commutative pseudo-ring with no zero divisors.

It can nicely be visualized as a commutative square:

This equation can conveniently be expressed by the commutative diagram

it can be checked that this is again an ultrafilter, and that the operation + is associative ( but not commutative ) on and extends the addition on ; 0 serves as a neutral element for the operation + on.