# Noncommutative rings and prime/maximal ideals

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 $R$?

• What does "(right, left) ideals of $R$" mean? You want us to choose either one? or you mean both, so that they are two-sided ideals? You should just say two-sided ideals, if you want to emphasize that. – rschwieb Aug 13 '18 at 20:14
• When you say "noncommutative" do you mean "not necessarily commutative" like everyone else? Or you want it to be not commutative? – rschwieb Aug 13 '18 at 20:15

## 1 Answer

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.

• Thanks! The same occurs with $\mahtbb{H}^{n} = \underbrace{\mathbb{H} \times \ldots \times \mathbb{H}_{n \ \mbox{times}}$? – 674123173797 - 4 Aug 17 '18 at 19:11
• @674123173797-4 Yes, it does. – rschwieb Aug 17 '18 at 20:19