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 rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields
Other articles related to "domain, principal ideal domain, principal, ideal domain, ideal, domains, principal ideal domains":
... 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 ...
... 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≤
... 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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