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 $M_n(k[(X_l{)}_{ij}\mid 1\le l\le m,1\le i,j\le n])$ 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 $k[(X_l{)}_{ij}{]}^{{\mathrm{GL}}_n(k)}$ since it is central and invariant.)

By definition, $F_n$ is a quotient of the free ring $k⟨t_1,\dots ,t_m⟩$ 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 $k⟨t_1,\dots ,t_m⟩$ 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 the Hilbert nullstellensatz or by the scheme theory). The free ring $k⟨t_1,\dots ,t_m⟩$ 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 ${\mathrm{Spec}}_n(A)$ 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 ${\mathrm{Spec}}_1(A)$ is the maximal spectrum of A and ${\mathrm{Spec}}_n(A)$ is empty for any $n>1$.

References

• Artin, Michael (1999). "Noncommutative Rings". http://math.mit.edu/~etingof/artinnotes.pdf. 
• Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6.