Prime Element iff Generates Principal Prime Ideal

Theorem

Integers

Let $\Z_{>0}$ be the set of strictly positive integers.

Let $p \in \Z_{>0}$.

Let $\ideal p$ be the principal ideal of $\Z$ generated by $p$.

Then $p$ is prime if and only if $\ideal p$ is a maximal ideal of $\Z$.

General Ring

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