Question

Give counter example to show that Every GCD (Greatest Common Divisor) Domain is not PID(Principal Ideal Domain)?

Answer 1

Answer on Question #50489, Math, Abstract Algebra

The most important examples of Euclidean domains are the ring Z of rational integers and the polynomial ring K[x] in one variable x over a field K.

The most counter examples were:

1) If D is a unique factorization domain, then the polynomial ring D[x] in one variable (and therefore in finitely many variables) over D is a unique factorization domain too

2) For D = 1;2;3;7, and 11, the ring R of algebraic integers in the field Q[√−D] is a Euclidean domain with the Euclidean norm φ, φ(a) = a¯a

3) The ring R of algebraic integers in the complex quadratic field Q[√−D] is not a Euclidean domain for D = 19;43;67, and 163.

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