Let $R = \mathbb{Z}[\sqrt 2]$ and $M_1,M_2 \subset R$ be maximal ideals.
Prove or disprove: If $R/M_1$ isomorphic to $R/M_2$ then $M_1=M_2$.
2026-03-26 19:18:31.1774552711
Are maximal Ideals in $\mathbb{Z}[\sqrt 2]$ equal when their quotient rings are isomorphic?
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Hint: Consider $M_1 = (3 - \sqrt{2})$ and $M_2=(3 + \sqrt{2})$.