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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

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Of coruse, if you mean left (right) ideals, then you are talking about a division ring. :) –  user641 Dec 2 '10 at 18:44

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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 $n \ge 2$.

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

http://en.wikipedia.org/wiki/Simple_ring

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).

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