In algebra, a generic matrix ring is a sort of a universal matrix ring.
Definition We denote by F n {\displaystyle F_{n}} a generic matrix ring of size n with variables X 1 , … X m {\displaystyle X_{1},\dots X_{m}} . It is characterized by the universal property: given a commutative ring R and n-by-n matrices A 1 , … , A m {\displaystyle A_{1},\dots ,A_{m}} over R, there exists a unique ring homomorphism (called the evaluation map) F n → M n ( R ) {\displaystyle F_{n}\to M_{n}(R)} extending the assignment X i ↦ A i {\displaystyle X_{i}\mapsto A_{i}} . Explicitly, given a field k, it is the subalgebra F n {\displaystyle F_{n}} of the matrix ring M n ( k [ ( X l ) i j ∣ 1 ≤ l ≤ m , 1 ≤ i , j ≤ n ] ) {\displaystyle M_{n}(k[(X_{l})_{ij}\mid 1\leq l\leq m,\ 1\leq i,j\leq n])} generated by n-by-n matrices X 1 , … , X m {\displaystyle X_{1},\dots ,X_{m}} , where ( X l ) i j {\displaystyle (X_{l})_{ij}} are matrix entries and commute by definition. For example, if m = 1 then F 1 {\displaystyle F_{1}} is a polynomial ring in one variable. For example, a central polynomial is an element of the ring F n {\displaystyle 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 ) {\displaystyle k[(X_{l})_{ij}]^{\operatorname {GL} _{n}(k)}} since it is central and invariant.) By definition, F n {\displaystyle F_{n}} is a quotient of the free ring k ⟨ t 1 , … , t m ⟩ {\displaystyle k\langle t_{1},\dots ,t_{m}\rangle } with t i ↦ X i {\displaystyle t_{i}\mapsto X_{i}} by the ideal consisting of all p that vanish identically on all n-by-n matrices over k.
Geometric perspective The universal property means that any ring homomorphism from k ⟨ t 1 , … , t m ⟩ {\displaystyle k\langle t_{1},\dots ,t_{m}\rangle } to a matrix ring factors through F n {\displaystyle F_{n}} . This has a following geometric meaning. In algebraic geometry, the polynomial ring k [ t , … , t m ] {\displaystyle k[t,\dots ,t_{m}]} is the coordinate ring of the affine space k m {\displaystyle k^{m}} , and to give a point of k m {\displaystyle k^{m}} is to give a ring homomorphism (evaluation) k [ t , … , t m ] → k {\displaystyle k[t,\dots ,t_{m}]\to k} (either by Hilbert's Nullstellensatz or by the scheme theory). The free ring k ⟨ t 1 , … , t m ⟩ {\displaystyle k\langle t_{1},\dots ,t_{m}\rangle } 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.)
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