Are there examples of non commutative rings which are not (which can not be) embedded in matrix rings?

Whatever example I know, all of them can be embedded in matrix ring. For example Quaternion ring, Group rings over a non-Abelian group.

  • $\begingroup$ Matrix ring over a commutative ring I assume? Do all your rings have a multiplicative identity? Do you allow matrices with an infinite number of entries? $\endgroup$ – arctic tern Aug 8 '16 at 6:36
  • $\begingroup$ Yes I mean over commutative ring. Yes as I am following the approach of Artin , I assume my ring must have identity (as a definition ). I would prefer a example where infinite entries are not allowed. But anyways I would be happy to know such examples too. $\endgroup$ – Tensor_Product Aug 8 '16 at 6:40
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    $\begingroup$ "Affine PI-algebras Not Embeddable in Matrix Rings" Ronald S. Irving, Journal of Algebra 82, 94-101 (1983). Have you access to this journal ? $\endgroup$ – Jean Marie Aug 8 '16 at 7:10
  • $\begingroup$ yes , I got it. Thank you very much. $\endgroup$ – Tensor_Product Aug 8 '16 at 7:13
  • $\begingroup$ A matrix ring $M_n(R)$ with $n>1$ apparently? Are you expecting $R$ to be a field or division ring, or anything else? $\endgroup$ – rschwieb Aug 8 '16 at 13:47

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