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.

Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one ; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals.

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 ).

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.

Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.

More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors ( e. g., Bourbaki ) refer to PIDs as principal rings.

Due to the characteristic entrywise procedure, this operation is identical to many multiplying ordinary numbers ( mn of them ) all at once-hence the Hadamard product is commutative, associative and distributive over addition, and is a principal submatrix of the Kronecker product.

* The Zariski-Samuel theorem determines the structure of a commutative principal ideal rings.

A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx ; or more generally an algebraic structure in which one of the principal binary operations is not commutative ; one also allows additional structures, e. g. topology or norm to be possibly carried by the noncommutative algebra of functions.

( If a quotient module R / I, for any commutative ring R and ideal I, is a projective R-module then I is principal.

Since every projective module over a principal ideal domain is free, one might conjecture that following is true: if R is a commutative ring such that every ( finitely generated ) projective R-module is free then every ( finitely generated ) projective R-module is free.

The principal novelty of the calculus of structures was its pervasive use of deep inference, which it was argued is necessary for calculi combining commutative and noncommutative operators ; this explanation concurs with the difficulty of designing sequent systems for pomset logic that have cut-elimination.

In commutative algebra and linear algebra, cyclic decomposition refers to writing a finitely generated module over a principal ideal domain as the direct sum of cyclic modules and one free module.

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

However there is a variation of Gauss's lemma that is valid even for polynomials over any commutative ring R, which replaces primitivity by the stronger property of co-maximality ( which is however equivalent to primitivity in the case of a Bézout domain, and in particular of a principal ideal domain ).

The prime ideals of the ring of integers are the ideals ( 0 ), ( 2 ), ( 3 ), ( 5 ), ( 7 ), ( 11 ), … The fundamental theorem of arithmetic generalizes to the Lasker – Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

An ideal P of a commutative ring R is prime if it has the following two properties:

* Nilradical of a ring, the nilradical of a commutative ring is a nilpotent ideal, which is as large as possible

When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

* If R is a unital commutative ring with an ideal m, then k = R / m is a field if and only if m is a maximal ideal.

* In a commutative ring with unity, every maximal ideal is a prime ideal.

* In commutative algebra, a commutative ring can be completed at an ideal ( in the topology defined by the powers of the ideal ).

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring.

For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.