In algebra, a generic matrix ring of size n with variables X 1 , … 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 n-by-n matrices A 1 , … , A m over R, any mapping X i ↦ A i extends to the ring homomorphism (called evaluation) F n → M n ( R ) .
Explicitly, given a field k, it is the subalgebra F n of the matrix ring M n ( k [ ( X l ) i j | 1 ≤ l ≤ m , 1 ≤ i , j ≤ n ] ) generated by n-by-n matrices X 1 , … , X m , where ( X l ) i j are matrix entries and commute by definition. For example, if m = 1, then F 1 is a polynomial ring in one variable.
For example, a central polynomial is an element of the ring F n that will map to a central element under an evaluation. (In fact, it is in the invariant ring k [ ( X l ) i j ] GL n ( k ) since it is central and invariant.)
By definition, F n is a quotient of the free ring k ⟨ t 1 , … , t m ⟩ with t i ↦ X i by the ideal consisting of all p that vanish identically on any n-by-n matrices over k. The universal property means that any ring homomorphism from k ⟨ t 1 , … , t m ⟩ to a matrix ring factors through F n . This has a following geometric meaning. In algebraic geometry, the polynomial ring k [ t , … , t m ] is the coordinate ring of the affine space k m and to give a point of k m is to give a ring homomorphism (evaluation) k [ t , … , t m ] → k (either by the Hilbert nullstellensatz or by the scheme theory). The free ring k ⟨ t 1 , … , t m ⟩ plays the role of the coordinate ring of the affine space in the noncommutative algebraic geometry (i.e., we don't demand free variables to commute) and thus a generic matrix ring of size n is the coordinate ring of a noncommutative affine variety whose points are the Spec's of matrix rings of size n (see below for a more concrete discussion.)
