Generic matrix ring

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

Definition

We denote by [math]\displaystyle{ F_n }[/math] a generic matrix ring of size n with variables [math]\displaystyle{ X_1, \dots X_m }[/math]. It is characterized by the universal property: given a commutative ring R and n-by-n matrices [math]\displaystyle{ A_1, \dots, A_m }[/math] over R, any mapping [math]\displaystyle{ X_i \mapsto A_i }[/math] extends to the ring homomorphism (called evaluation) [math]\displaystyle{ F_n \to M_n(R) }[/math].

Explicitly, given a field k, it is the subalgebra [math]\displaystyle{ F_n }[/math] of the matrix ring [math]\displaystyle{ M_n(k[(X_l)_{ij} \mid 1 \le l \le m,\ 1 \le i, j \le n]) }[/math] generated by n-by-n matrices [math]\displaystyle{ X_1, \dots, X_m }[/math], where [math]\displaystyle{ (X_l)_{ij} }[/math] are matrix entries and commute by definition. For example, if m = 1 then [math]\displaystyle{ F_1 }[/math] is a polynomial ring in one variable.

For example, a central polynomial is an element of the ring [math]\displaystyle{ F_n }[/math] that will map to a central element under an evaluation. (In fact, it is in the invariant ring [math]\displaystyle{ k[(X_l)_{ij}]^{\operatorname{GL}_n(k)} }[/math] since it is central and invariant.)

By definition, [math]\displaystyle{ F_n }[/math] is a quotient of the free ring [math]\displaystyle{ k\langle t_1, \dots, t_m \rangle }[/math] with [math]\displaystyle{ t_i \mapsto X_i }[/math] 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 [math]\displaystyle{ k\langle t_1, \dots, t_m \rangle }[/math] to a matrix ring factors through [math]\displaystyle{ F_n }[/math]. This has a following geometric meaning. In algebraic geometry, the polynomial ring [math]\displaystyle{ k[t, \dots, t_m] }[/math] is the coordinate ring of the affine space [math]\displaystyle{ k^m }[/math], and to give a point of [math]\displaystyle{ k^m }[/math] is to give a ring homomorphism (evaluation) [math]\displaystyle{ k[t, \dots, t_m] \to k }[/math] (either by the Hilbert nullstellensatz or by the scheme theory). The free ring [math]\displaystyle{ k\langle t_1, \dots, t_m \rangle }[/math] 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 [math]\displaystyle{ \operatorname{Spec}_n(A) }[/math] denote the set of all maximal ideals [math]\displaystyle{ \mathfrak{m} }[/math] in A such that [math]\displaystyle{ A/\mathfrak{m} \approx M_n(k) }[/math]. If A is commutative, then [math]\displaystyle{ \operatorname{Spec}_1(A) }[/math] is the maximal spectrum of A and [math]\displaystyle{ \operatorname{Spec}_n(A) }[/math] is empty for any [math]\displaystyle{ n \gt 1 }[/math].

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.