Generic matrix ring

In algebra, a generic matrix ring is a sort of a universal matrix ring.
Definition.
We denote by formula_1 a generic matrix ring of size "n" with variables formula_2. It is characterized by the universal property: given a commutative ring "R" and "n"-by-"n" matrices formula_3 over "R", any mapping formula_4 extends to the ring homomorphism (called evaluation) formula_5.
Explicitly, given a field "k", it is the subalgebra formula_1 of the matrix ring formula_7 generated by "n"-by-"n" matrices formula_8, where formula_9 are matrix entries and commute by definition. For example, if "m" = 1 then formula_10 is a polynomial ring in one variable.
For example, a central polynomial is an element of the ring formula_1 that will map to a central element under an evaluation. (In fact, it is in the invariant ring formula_12 since it is central and invariant.)
By definition, formula_1 is a quotient of the free ring formula_14 with formula_15 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 formula_14 to a matrix ring factors through formula_1. This has a following geometric meaning. In algebraic geometry, the polynomial ring formula_18 is the coordinate ring of the affine space formula_19, and to give a point of formula_19 is to give a ring homomorphism (evaluation) formula_21 (either by the Hilbert nullstellensatz or by the scheme theory). The free ring formula_14 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 formula_23 denote the set of all maximal ideals formula_24 in "A" such that formula_25. If "A" is commutative, then formula_26 is the maximal spectrum of "A" and formula_23 is empty for any formula_28.