Example of a non-commutative rings with identity that do not contain non-trival ideals and are not division rings 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

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).
