Generic matrix ring

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

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

extends to the ring homomorphism (called evaluation)

Explicitly, given a field k, it is the subalgebra

of the matrix ring

generated by n-by-n matrices

are matrix entries and commute by definition.

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.

(In fact, it is in the invariant ring

{\displaystyle k[(X_{l})_{ij}]^{\operatorname {GL} _{n}(k)}}

since it is central and invariant.

is a quotient of the free ring

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 (evaluation)

(either by Hilbert's Nullstellensatz or by the scheme theory).

The free ring

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.)

For simplicity, assume k is algebraically closed.

Let A be an algebra over k and let

denote the set of all maximal ideals

If A is commutative, then

is the maximal spectrum of A and