Generic matrix ring

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

Definition

We denote by ${\displaystyle F_{n}}$ a generic matrix ring of size n with variables ${\displaystyle X_{1},\dots X_{m}}$. It is characterized by the universal property: given a commutative ring R and n-by-n matrices ${\displaystyle A_{1},\dots ,A_{m}}$ over R, any mapping ${\displaystyle X_{i}\mapsto A_{i}}$ extends to the ring homomorphism (called evaluation) ${\displaystyle F_{n}\to M_{n}(R)}$.

Explicitly, given a field k, it is the subalgebra ${\displaystyle F_{n}}$ of the matrix ring ${\displaystyle M_{n}(k[(X_{l})_{ij}\mid 1\leq l\leq m,\ 1\leq i,j\leq n])}$ generated by n-by-n matrices ${\displaystyle X_{1},\dots ,X_{m}}$, where ${\displaystyle (X_{l})_{ij}}$ are matrix entries and commute by definition. For example, if m = 1 then ${\displaystyle F_{1}}$ is a polynomial ring in one variable.

For example, a central polynomial is an element of the ring ${\displaystyle F_{n}}$ 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.[1])

By definition, ${\displaystyle F_{n}}$ is a quotient of the free ring ${\displaystyle k\langle t_{1},\dots ,t_{m}\rangle }$ with ${\displaystyle t_{i}\mapsto X_{i}}$ 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 ${\displaystyle k\langle t_{1},\dots ,t_{m}\rangle }$ to a matrix ring factors through ${\displaystyle F_{n}}$. This has a following geometric meaning. In algebraic geometry, the polynomial ring ${\displaystyle k[t,\dots ,t_{m}]}$ is the coordinate ring of the affine space ${\displaystyle k^{m}}$, and to give a point of ${\displaystyle k^{m}}$ is to give a ring homomorphism (evaluation) ${\displaystyle k[t,\dots ,t_{m}]\to k}$ (either by the Hilbert nullstellensatz or by the scheme theory). The free ring ${\displaystyle k\langle t_{1},\dots ,t_{m}\rangle }$ 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 ${\displaystyle \operatorname {Spec} _{n}(A)}$ denote the set of all maximal ideals ${\displaystyle {\mathfrak {m}}}$ in A such that ${\displaystyle A/{\mathfrak {m}}\approx M_{n}(k)}$. If A is commutative, then ${\displaystyle \operatorname {Spec} _{1}(A)}$ is the maximal spectrum of A and ${\displaystyle \operatorname {Spec} _{n}(A)}$ is empty for any ${\displaystyle n>1}$.

References

1. Artin 1999, Proposition V.15.2.

• Artin, Michael (1999). "Noncommutative Rings" (PDF).
• Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001.