Principal Ideal Domain - Examples

Examples

Examples include:

  • K: any field,
  • Z: the ring of integers,
  • K: rings of polynomials in one variable with coefficients in a field. (The converse is also true; that is, if A is a PID, then A 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 .
  • Z: the ring of Gaussian integers
  • Z (where ω is a primitive cube root of 1): the Eisenstein integers

Examples of integral domains that are not PIDs:

  • Z: 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: The ideal (x,y) is not principal.

Read more about this topic:  Principal Ideal Domain

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