Generic matrix ring explained

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,} ... X _{m} . It is characterized by the universal property: given a commutative ring R and n-by-n matrices A _{1,} ... , A _{m} over R, any mapping X _{i} \mapsto A _{i} extends to the ring homomorphism (called evaluation) F _{n} \to M _{n(R)} .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,} ... , X _{m} , where (X _{l)} _{ij} are matrix entries and commute by definition. For example, if m = 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,} ... , 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,} ... , 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, ... , 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, ... , t _{m]} \to k (either by Hilbert's Nullstellensatz or by the scheme theory). The free ring k\langle t _{1,} ... , 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 ak{m} in A such that A/ ak{m} ≈ 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 .

References

• Web site: Artin. Michael. Noncommutative Rings. 1999.
• Book: Cohn, Paul M. . Paul Cohn

. Paul Cohn . Revised ed. of Algebra, 2nd . Further algebra and applications . 2003 . London . . 1-85233-667-6 . 1006.00001 .