A noncommutative Hilbert basis theorem and subrings of matrices

A finitely generated central extension $A[{u_1}, \ldots ,{u_k}]$ of a commutative noetherian ring A, satisfies the ascending chain condition for ideals P for which $A[{u_1}, \ldots ,{u_k}]/P$ can be embedded in matrix rings ${M_n}(K)$ over arbitrary commutative rings K and n bounded. The method of proof leads to an example of a ring R which satisfies the same identities of ${M_n}(K)$ but nevertheless cannot be embedded in any matrix ring over a commutative ring of arbitrary finite order.