Generic matrix ring

In algebra, a generic matrix ring of size n with matrix variables $X_{1},\dots X_{m}$, denoted by $F_{n}$, is a sort of a universal matrix ring. It is universal in the sense that, given a commutative ring R and matrices $A_{1},\dots ,A_{m}$ over R, any mapping $X_{i}\mapsto A_{i}$ extends to the ring homomorphism (called evaluation) $F_{n}\to M_{n}(R)$. For example, a central polynomial is an element of the ring $F_{n}$.

Explicitly, it is the subalgebra $F_{n}$ of the matrix ring $M_{n}(k[(X_{l})_{ij}|1\leq l\leq m,1\leq i,j\leq n])$ generated by n-by-n matrices $X_{1},\dots ,X_{m}$, where $(X_{l})_{ij}$ are matrix entries and commute by definition. For example, if m = 1, then $F_{1}$ is a polynomial ring in one variable.

By definition, $F_{n}$ is a quotient of the free ring $k\langle t_{1},\dots ,t_{m}\rangle$.

References

• Artin, Michael (1999). "Noncommutative Rings" (PDF).
• Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001.