Suppose that $R$ is a commutative unital ring containing two non-units $r$ and $s$ satisfying $r+s=1$. I am trying to show that $(r)$ and $(s)$ form distinct maximal ideals in $R$. That they are distinct is obvious, but I am having trouble showing they are maximal ideals. I could use a hint.
2026-04-02 10:25:03.1775125503
Distinct, Nonunit Elements Form Two Distinct Maximal Ideals
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I don't think this is true. Consider $R=\mathbb{Z}$, $r=6$, and $s=-5$. Then $(r)$ is not maximal since $(6) \subseteq (2) \subseteq \mathbb{Z}$ with both inclusions being proper.