Editing Integral domain (section)

== Examples ==
* The archetypical example is the ring <math>\Z</math> of all [[integer]]s.
* Every [[field (mathematics)|field]] is an integral domain. For example, the field <math>\R</math> of all [[real number]]s is an integral domain.  Conversely, every [[artinian ring|Artinian]] integral domain is a field. In particular, all finite integral domains are [[finite field]]s (more generally, by [[Wedderburn's little theorem]], finite [[Domain (ring theory)|domains]] are [[finite field]]s). The ring of integers <math>\Z</math> provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:
*: <math>\Z \supset 2\Z \supset \cdots \supset 2^n\Z \supset 2^{n+1}\Z \supset \cdots</math>
* Rings of [[polynomial]]s are integral domains if the coefficients come from an integral domain. For instance, the ring <math>\Z[x]</math> of all polynomials in one variable with integer coefficients is an integral domain; so is the ring <math>\Complex[x_1,\ldots,x_n]</math> of all polynomials in ''n''-variables with [[Complex number|complex]] coefficients.
* The previous example can be further exploited by taking quotients from prime ideals. For example, the ring <math>\Complex[x,y]/(y^2 - x(x-1)(x-2))</math> corresponding to a plane [[elliptic curve]] is an integral domain. Integrality can be checked by showing <math>y^2 - x(x-1)(x-2)</math> is an [[irreducible polynomial]].
* The ring <math>\Z[x]/(x^2 - n) \cong \Z[\sqrt{n}]</math> is an integral domain for any non-square integer <math>n</math>. If <math>n > 0</math>, then this ring is always a subring of <math>\R</math>, otherwise, it is a subring of <math>\Complex.</math>
* The ring of [[p-adic number|''p''-adic integers]] <math>\Z_p</math> is an integral domain.
* The ring of [[formal power series]] of an integral domain is an integral domain.
* If <math>U</math> is a [[connectedness|connected]] [[open subset]] of the [[complex number|complex plane]] <math>\Complex</math>, then the ring <math>\mathcal{H}(U)</math> consisting of all [[holomorphic function]]s is an integral domain. The same is true for rings of [[analytic function]]s on connected open subsets of analytic [[manifold]]s.
* A [[regular local ring]] is an integral domain. In fact, a regular local ring is a [[unique factorization domain|UFD]].{{sfn|Auslander|Buchsbaum|1959|ps=none}}{{sfn|Nagata|1958|ps=none}}