Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm looking for an example of a non-commutative ring, $R$, with identity s.t $R$ does not contain a non-trival 2 sided ideal and $R$ is not a division ring

share|cite|improve this question
Of coruse, if you mean left (right) ideals, then you are talking about a division ring. :) – user641 Dec 2 '10 at 18:44
up vote 9 down vote accepted

If "ideal" means "two-sided ideal," then the Weyl algebra $\mathbb{C} \langle x, y \rangle / (xy - yx - 1)$ is an example. So is $\mathcal{M}_n(R)$ for any division ring $R$ and $n \ge 2$.

share|cite|improve this answer

If you mean two-sided ideals, you are looking for simple rings (that are not division rings):

E.g. as in the wikipedia article, is the ring of matrices (of a certain size) over a field. Clearly this is not a division ring since not every matrix is invertible (matrices with zero determinants are not invertible).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.