In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.^[1][2] Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity.^[3][4] Noncommutative integral domains are sometimes admitted.^[5] This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings.

Some sources, notably Lang, use the term entire ring for integral domain.^[6]

Some specific kinds of integral domains are given with the following chain of class inclusions:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

Definition

An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Equivalently:

• An integral domain is a nonzero commutative ring with no nonzero zero divisors.
• An integral domain is a commutative ring in which the zero ideal {0} is a prime ideal.
• An integral domain is a nonzero commutative ring for which every nonzero element is cancellable under multiplication.
• An integral domain is a ring for which the set of nonzero elements is a commutative monoid under multiplication (because a monoid must be closed under multiplication).
• An integral domain is a nonzero commutative ring in which for every nonzero element r, the function that maps each element x of the ring to the product xr is injective. Elements r with this property are called regular, so it is equivalent to require that every nonzero element of the ring be regular.
• An integral domain is a ring that is isomorphic to a subring of a field. (Given an integral domain, one can embed it in its field of fractions.)

Examples

• The archetypical example is the ring ${\displaystyle \mathbb {Z} }$ of all integers.
• Every field is an integral domain. For example, the field ${\displaystyle \mathbb {R} }$ of all real numbers is an integral domain. Conversely, every Artinian integral domain is a field. In particular, all finite integral domains are finite fields (more generally, by Wedderburn's little theorem, finite domains are finite fields). The ring of integers ${\displaystyle \mathbb {Z} }$ provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:

  ${\displaystyle \mathbb {Z} \supset 2\mathbb {Z} \supset \cdots \supset 2^{n}\mathbb {Z} \supset 2^{n+1}\mathbb {Z} \supset \cdots }$

• Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring ${\displaystyle \mathbb {Z} }$ of all polynomials in one variable with integer coefficients is an integral domain; so is the ring ${\displaystyle \mathbb {C} }$ of all polynomials in n-variables with complex coefficients.
• The previous example can be further exploited by taking quotients from prime ideals. For example, the ring ${\displaystyle \mathbb {C} /(y^{2}-x(x-1)(x-2))}$corresponding to a plane elliptic curve is an integral domain. Integrality can be checked by showing ${\displaystyle y^{2}-x(x-1)(x-2)}$is an irreducible polynomial.
• The ring ${\displaystyle \mathbb {Z} /(x^{2}-n)\cong \mathbb {Z} {\sqrt {n}}}$ is an integral domain for any non-square integer ${\displaystyle n}$. If ${\displaystyle n>0}$, then this ring is always a subring of ${\displaystyle \mathbb {R} }$, otherwise, it is a subring of ${\displaystyle \mathbb {C} .}$
• The ring of p-adic integers ${\displaystyle \mathbb {Z} _{p}}$ is an integral domain.
• The ring of formal power series of an integral domain is an integral domain.
• If ${\displaystyle U}$ is a connected open subset of the complex plane ${\displaystyle \mathbb {C} }$, then the ring ${\displaystyle {\mathcal {H}}(U)}$ consisting of all holomorphic functions is an integral domain. The same is true for rings of analytic functions on connected open subsets of analytic manifolds.
• A regular local ring is an integral domain. In fact, a regular local ring is a UFD.^[7][8]

Non-examplesedit

The following rings are not integral domains.

• The zero ring (the ring in which ${\displaystyle 0=1}$).
• The quotient ring ${\displaystyle \mathbb{Z}/m\mathbb{Z}}$Zdroj:https://en.wikipedia.org?pojem=Commutative_domain
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