Generic matrix

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

We denote by ${\displaystyle F_{n}}$ a generic matrix ring of size n with variables ${\displaystyle X_{1},\dots X_{m}}$. It is characterized by the universal property: given a commutative ring R and n-by-n matrices ${\displaystyle A_{1},\dots ,A_{m}}$ over R, any mapping ${\displaystyle X_{i}\mapsto A_{i}}$ extends to the ring homomorphism (called evaluation) ${\displaystyle F_{n}\to M_{n}(R)}$.

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