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Principal Ideal Domain


A principal ideal domain is an integral domain in which every proper ideal can be generated by a single element. The term "principal ideal domain" is often abbreviated P.I.D. Examples of P.I.D.s include the integers, the Gaussian integers, and the set of polynomials in one variable with real coefficients.

Every Euclidean ring is a principal ideal domain, but the converse is not true. Nevertheless, the notion of greatest common divisor arising from the Euclidean algorithm can be extended to the more general context of principal ideal domains as follows. Given two nonzero elements a,b of a principal ideal domain R, a greatest common divisor of a and b is defined as any element d of R such that

 <a,b>=<d>.

Every principal ideal domain is a unique factorization domain, but not conversely. Every polynomial ring over a field is a unique factorization domain, but it is a principal ideal domain iff the number of indeterminates is one.


See also

Algebraic Number Theory, Krull's Principal Ideal Theorem, Principal Ring

Portions of this entry contributed by Margherita Barile

Portions of this entry contributed by John Renze

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Cite this as:

Barile, Margherita; Renze, John; and Weisstein, Eric W. "Principal Ideal Domain." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrincipalIdealDomain.html

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