Principal ideal domain

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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.

Contents

Examples

Examples include:

  • K: any field,
  • Z: the ring of integers[1],
  • 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[2]
  • 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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