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

Suppose $\displaystyle A$ is an ideal in a commutative ring $\displaystyle R$ with identity. Show that if $\displaystyle cA$ is principal for some element $\displaystyle c$, then $\displaystyle A$ 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 $\displaystyle x$ so that $\displaystyle A$ is generated by it. I started with an example, taking $\displaystyle R=\mathbb{Z}$ and the ideal to be $\displaystyle A=2\mathbb{Z}$. I multiplied it by 10: $\displaystyle 10(2\mathbb{Z})$, which turns out to be $\displaystyle 20\mathbb{Z}$. Now, in order to recover the generator for $\displaystyle A$, 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!