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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