Principal ideal is isomorphic to integral domain as a module

Verbal statement
In an integral domain, any principal ideal is isomorphic, as a module, to the whole ring.

Symbolic statement
Let $$A$$ be an integral domain and $$I$$ a principal ideal in $$A$$, generated by an element $$x \in A$$. Then $$I$$ is isomorphic to $$A$$ as an $$A$$-module.

Applications

 * Ideal in integral domain implies self-similar