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Concept
Principal ideal domain
Summary
In mathematics, 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 that 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 domains. All Euclidean domains and all fields are principal ideal domains.
Principal ideal domains appear in the following chain of class inclusions:
Examples include:
any field,
the ring of integers,
rings of polynomials in one variable with coefficients in a field. (The converse is also true, i.e. if is a PID then is a field.) Furthermore, a ring of formal power series in one variable over a field is a PID since every ideal is of the form ,
the ring of Gaussian integers,
(where is a primitive cube root of 1): the Eisenstein integers,
Any discrete valuation ring, for instance the ring of p-adic integers .
Examples of integral domains that are not PIDs:
is an example of a ring which is not a unique factorization domain, since Hence it is not a principal ideal domain because principal ideal domains are unique factorization domains. Also, is an ideal that cannot be generated by a single element.
the ring of all polynomials with integer coefficients.
Official source
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