Principal ideal is isomorphic to integral domain as a module

This article is about the statement of a simple but indispensable lemma in commutative algebra
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Statement

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