Generic matrix ring

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

Definition

We denote by a generic matrix ring of size n with variables. It is characterized by the universal property: given a commutative ring R and n-by-n matrices over R, there exists a unique ring homomorphism extending the assignment.
Explicitly, given a field k, it is the subalgebra of the matrix ring generated by n-by-n matrices, where are matrix entries and commute by definition. For example, if m = 1 then is a polynomial ring in one variable.
For example, a central polynomial is an element of the ring that will map to a central element under an evaluation.
By definition, is a quotient of the free ring with by the ideal consisting of all p that vanish identically on all n-by-n matrices over k.

The universal property means that any ring homomorphism from to a matrix ring factors through. This has a following geometric meaning. In algebraic geometry, the polynomial ring is the coordinate ring of the affine space, and to give a point of is to give a ring homomorphism . The free ring plays the role of the coordinate ring of the affine space in the noncommutative algebraic geometry 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''

For simplicity, assume k is algebraically closed. Let A be an algebra over k and let denote the set of all maximal ideals in A such that. If A is commutative, then is the maximal spectrum of A and is empty for any.