Hello everyone. I have been trying to work out the following problem:

Suppose is an ideal in a commutative ring with identity. Show that if is principal for some element , then is principal.

This seems like a very simple and innocuous statement, but I can't seem to prove it. My idea is to try to construct the element so that is generated by it. I started with an example, taking and the ideal to be . I multiplied it by 10: , which turns out to be . Now, in order to recover the generator for , I need to divide 20 by 10. But this is a serious issue for the general statement above, because I am not guaranteed any kind of division or cancellation. It is completely unclear to me how to pick up the single generator.

I would appreciate any input anybody has on this problem!