For example, I kind of expect that in $\mathbb{Z}_{6}$ $\overline{2}$ is prime but I don't know how to prove it (considering the fact that I can test all the different products, but I want a more theorical proof), because I want to show that $\left(\overline{2}\right)$ is a prime ideal. Thanks in advance!
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One thing you could do is show that the quotient by said ideal forms an integral domain, and so it must be a prime ideal (as $R/I$ is an integral domain $\iff I$ prime, and if $I$ is prime and generated by $i$, then $i$ is prime) – Theo C. Mar 11 '19 at 15:43
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In $\mathbb{Z}_{n}$ $\left(p\right)$ is a prime ideal $\Leftrightarrow$ $p$ is prime? – Jack Talion Mar 11 '19 at 15:48