Generic matrix ring

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

Definition
We denote by $$F_n$$ a generic matrix ring of size n with variables $$X_1, \dots X_m$$. It is characterized by the universal property: given a commutative ring R and n-by-n matrices $$A_1, \dots, A_m$$ over R, any mapping $$X_i \mapsto A_i$$ extends to the ring homomorphism (called evaluation) $$F_n \to M_n(R)$$.

Explicitly, given a field k, it is the subalgebra $$F_n$$ of the matrix ring $$M_n(k[(X_l)_{ij} \mid 1 \le l \le m,\ 1 \le i, j \le n])$$ 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&thinsp;=&thinsp;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 $$k[(X_l)_{ij}]^{\operatorname{GL}_n(k)}$$ since it is central and invariant. )

By definition, $$F_n$$ is a quotient of the free ring $$k\langle t_1, \dots, t_m \rangle$$ with $$t_i \mapsto X_i$$ 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 $$k\langle t_1, \dots, t_m \rangle$$ to a matrix ring factors through $$F_n$$. This has a following geometric meaning. In algebraic geometry, the polynomial ring $$k[t, \dots, t_m]$$ 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) $$k[t, \dots, t_m] \to k$$ (either by Hilbert's Nullstellensatz or by the scheme theory). The free ring $$k\langle t_1, \dots, t_m \rangle$$ 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.)

The maximal spectrum of a generic matrix ring
For simplicity, assume k is algebraically closed. Let A be an algebra over k and let $$\operatorname{Spec}_n(A)$$ denote the set of all maximal ideals $$\mathfrak{m}$$ in A such that $$A/\mathfrak{m} \approx M_n(k)$$. If A is commutative, then $$\operatorname{Spec}_1(A)$$ is the maximal spectrum of A and $$\operatorname{Spec}_n(A)$$ is empty for any $$n > 1$$.