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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