## Over ZZ, if (a,b) is maximal then gcd(a,b) is prime

Let $a,b \in \mathbb{Z}$ such that $(a,b)$ is maximal as an ideal in $\mathbb{Z}$. What can be said about $a$ and $b$?

Since $\mathbb{Z}$ is a principal ideal domain, $(a,b) = (d)$ for some $d$. If $(d)$ is maximal, then it is prime, so that $d$ is a prime element. Moreover, $d$ is a greatest common divisor of $a$ and $b$. So $\mathsf{gcd}(a,b)$ is prime.

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