Editing Commutative algebra (section)

==Overview==
Commutative algebra is essentially the study of the rings occurring in [[algebraic number theory]] and [[algebraic geometry]].

Several concepts of commutative algebras have been developed in relation with algebraic number theory, such as [[Dedekind ring]]s (the main class of commutative rings occurring in algebraic number theory), [[integral extension]]s, and [[valuation ring]]s.

[[Polynomial ring]]s in several indeterminates over a [[field (mathematics)|field]] are examples of commutative rings. Since algebraic geometry is fundamentally the study of the common [[zero of a function|zeros]] of these rings, many results and concepts of algebraic geometry have counterparts in commutative algebra, and their names recall often their geometric origin; for example "[[Krull dimension]]", "[[localization of a ring]]", "[[local ring]]", "[[regular ring]]".

An [[affine algebraic variety]] corresponds to a [[prime ideal]] in a polynomial ring, and the points of such an affine variety correspond to the [[maximal ideal]]s that contain this prime ideal. The [[Zariski topology]], originally defined on an algebraic variety, has been extended to the sets of the prime ideals of any commutative ring; for this topology, the [[closed set]]s are the sets of prime ideals that contain a given ideal. 

The [[spectrum of a ring]] is a [[ringed space]] formed by the prime ideals equipped with the Zariski topology, and the localizations of the ring at the [[open set]]s of a [[basis (topology)|basis]] of this topology. This is the starting point of [[scheme theory]], a generalization of algebraic geometry introduced by [[Grothendieck]], which is strongly based on commutative algebra, and has induced, in turns, many developments of commutative algebra.