# Non Commutative rings which are not embeddable in matrix rings

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.

• 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? – arctic tern Aug 8 '16 at 6:36
• 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. – Tensor_Product Aug 8 '16 at 6:40
• "Affine PI-algebras Not Embeddable in Matrix Rings" Ronald S. Irving, Journal of Algebra 82, 94-101 (1983). Have you access to this journal ? – Jean Marie Aug 8 '16 at 7:10
• yes , I got it. Thank you very much. – Tensor_Product Aug 8 '16 at 7:13
• A matrix ring $M_n(R)$ with $n>1$ apparently? Are you expecting $R$ to be a field or division ring, or anything else? – rschwieb Aug 8 '16 at 13:47