# Principal Ideal Domain

## Homework Statement

Prove that R is a PID (principal ideal domain) when R is a ring such that Z $$\subset$$ R $$\subset$$ Q (Z=integers, Q=rationals)

## The Attempt at a Solution

So I'm not really sure how to start this problem. I know that a principal ideal domain is an integral domain in which every ideal is principal. Z is euclidean domain and therefore a PID, Q is a field and therefore a PID. So R is 'between' 2 PIDs. Also if I could show it was a Euclidean Domain then it would be a PID.

Thanks!

## Answers and Replies

Try to picture explicitly what a subring of $$\mathbb{Q}$$ might look like. Take $$\mathbb{Z}$$ and throw in a fraction or two; what does that generate?

Hurkyl
Staff Emeritus