Generic matrix ring

In algebra, a generic matrix ring is a sort of a universal matrix ring.

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 formal 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.

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.