Let $R$ a non-simple noncommutative ring, and let $\mathcal{I}$ the set of non-trivial (right, left) ideals of $R$, with the following property: "Every element $I \in \mathcal{I}$ is prime and/or maximal". There exists an example of such $R$?
2026-03-25 07:41:54.1774424514
Noncommutative rings and prime/maximal ideals
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Let $R=\mathbb H\times \mathbb H$ where $\mathbb H$ denotes Hamilton's quaternions.
It has four right ideals, all of which are two sided, two of which are trivial, and the other two are maximal and prime.