# Principal Ideal Domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. 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. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax + by.

Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind rings. All Euclidean domains and all fields are principal ideal domains.

Commutative ringsintegral domainsintegrally closed domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfields

### Other articles related to "domain, principal ideal domain, principal, ideal domain, ideal, domains, principal ideal domains":

Free Ideal Ring - Properties and Examples
... It turns out that a left and right fir is a domain ... Furthermore, a commutative fir is precisely a principal ideal domain, while a commutative semifir is precisely a Bézout domain ... Every principal right ideal domain R is a right fir, since every nonzero principal right ideal of a domain is isomorphic to R ...
Principal Ideal Domain - Properties
... In a principal ideal domain, any two elements a,b have a greatest common divisor, which may be obtained as a generator of the ideal (a,b) ... All Euclidean domains are principal ideal domains, but the converse is not true ... An example of a principal ideal domain that is not a Euclidean domain is the ring In this domain no q and r exist, with 0≤
Discrete Valuation Ring
... In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal ... This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions R is a local principal ideal domain, and not a field ... R is a local Dedekind domain and not a field ...

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