Generic matrix ring

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

Definition

We denote by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_n} a generic matrix ring of size n with variables Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_l)_{ij}} are matrix entries and commute by definition. For example, if m = 1 then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_1} is a polynomial ring in one variable.

For example, a central polynomial is an element of the ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_n} that will map to a central element under an evaluation. (In fact, it is in the invariant ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k[(X_l)_{ij}]^{\operatorname{GL}_n(k)}} since it is central and invariant.)

By definition, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_n} is a quotient of the free ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\langle t_1, \dots, t_m \rangle} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\langle t_1, \dots, t_m \rangle} to a matrix ring factors through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_n} . This has a following geometric meaning. In algebraic geometry, the polynomial ring Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k[t,\dots ,t_{m}]} is the coordinate ring of the affine space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^m} , and to give a point of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^m} is to give a ring homomorphism (evaluation) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k[t, \dots, t_m] \to k} (either by Hilbert's Nullstellensatz or by the scheme theory). The free ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Spec}_n(A)} denote the set of all maximal ideals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{m}} in A such that $A/{\mathfrak {m}}\approx M_{n}(k)$. If A is commutative, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Spec}_1(A)} is the maximal spectrum of A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Spec}_n(A)} is empty for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.