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.
- K: any field,
- Z: the ring of integers,
- K[x]: rings of polynomials in one variable with coefficients in a field. (The converse is also true; that is, if A[x] is a PID, then A is a field.) Furthermore, a ring of formal power series over a field is a PID since every ideal is of the form (xk).
- Z[i]: the ring of Gaussian integers
- Z[ω] (where ω is a primitive cube root of 1): the Eisenstein integers
Examples of integral domains that are not PIDs:
- Z[x]: the ring of all polynomials with integer coefficients --- it is not principal because the ideal generated by 2 and X is an example of an ideal that cannot be generated by a single polynomial.
- K[x,y]: The ideal (x,y) is not principal.
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