## Rings of fractions over a PID are PIDs

Prove that if is a principal ideal domain and a set of denominators, then the set of fractions with denominators in is a principal ideal domain.

is a subring of a field (the field of fractions of ), and thus is an integral domain.

Let be an ideal. Let . We claim that is an ideal of . To see this, note that , so that . Now let ; then , so that . Finally, let and , with . Then , so that . Thus is an ideal of , and thus we have . Let such that ; then . Now let with ; we have , so that is principal.

Thus is a principal ideal domain.

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