Generic matrix ring

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

Explicitly, given a field k, it is the subalgebra $F_{n}$ of the matrix ring $M_{n}(k[(X_{l})_{ij}\mid 1\leq l\leq m,\ 1\leq i,j\leq n])$ generated by n-by-n matrices $X_{1},\dots ,X_{m}$, 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 $k\langle t_{1},\dots ,t_{m}\rangle$ 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 $k\langle t_{1},\dots ,t_{m}\rangle$ to a matrix ring factors through $F_{n}$. This has a following geometric meaning. In algebraic geometry, the polynomial ring $k[t,\dots ,t_{m}]$ 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) $k[t,\dots ,t_{m}]\to k$ (either by Hilbert's Nullstellensatz or by the scheme theory). The free ring $k\langle t_{1},\dots ,t_{m}\rangle$ 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 $\operatorname {Spec} _{n}(A)$ denote the set of all maximal ideals ${\mathfrak {m}}$ in A such that $A/{\mathfrak {m}}\approx M_{n}(k)$. 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

• 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.