Generic matrix ring

We denote by $F\_n$ a generic matrix ring of size n with variables $X\_1,\; \backslash dots\; X\_m$. It is characterized by the universal property: given a commutative ring R and n-by-n matrices $A\_1,\; \backslash dots,\; A\_m$ over R, any mapping $X\_i\; \backslash mapsto\; A\_i$ extends to the ring homomorphism (called evaluation) $F\_n\; \backslash to\; M\_n(R)$.

Explicitly, given a field k, it is the subalgebra $F\_n$ of the matrix ring $M\_n(k[(X\_l)\_\{ij\}\; \backslash mid\; 1\; \backslash le\; l\; \backslash le\; m,\backslash \; 1\; \backslash le\; i,\; j\; \backslash le\; n])$ generated by n-by-n matrices $X\_1,\; \backslash dots,\; X\_m$, 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 $k[(X\_l)\_\{ij\}]^\{\backslash operatorname\{GL\}\_n(k)\}$ since it is central and invariant.{{harvnb|Artin|1999|loc=Proposition V.15.2.}})

By definition, $F\_n$ is a quotient of the free ring $k\backslash langle\; t\_1,\; \backslash dots,\; t\_m\; \backslash rangle$ with $t\_i\; \backslash mapsto\; X\_i$ by the ideal consisting of all p that vanish identically on all n-by-n matrices over k.

The universal property means that any ring homomorphism from $k\backslash langle\; t\_1,\; \backslash dots,\; t\_m\; \backslash rangle$ to a matrix ring factors through $F\_n$. This has a following geometric meaning. In algebraic geometry, the polynomial ring $k[t,\; \backslash dots,\; t\_m]$ 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) $k[t,\; \backslash dots,\; t\_m]\; \backslash to\; k$ (either by Hilbert's Nullstellensatz or by the scheme theory). The free ring $k\backslash langle\; t\_1,\; \backslash dots,\; t\_m\; \backslash rangle$ 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.)

{{expand section|date=June 2014}}

For simplicity, assume k is algebraically closed. Let A be an algebra over k and let $\backslash operatorname\{Spec\}\_n(A)$ denote the set of all maximal ideals $\backslash mathfrak\{m\}$ in A such that $A/\backslash mathfrak\{m\}\; \backslash approx\; M\_n(k)$. If A is commutative, then $\backslash operatorname\{Spec\}\_1(A)$ is the maximal spectrum of A and $\backslash operatorname\{Spec\}\_n(A)$ is empty for any $n\; >\; 1$.

{{reflist}}

• {{cite web|last=Artin|first=Michael|title=Noncommutative Rings|url=http://math.mit.edu/~etingof/artinnotes.pdf|year=1999}}
• {{cite book | first=Paul M. | last=Cohn | authorlink=Paul Cohn | edition=Revised ed. of Algebra, 2nd | title=Further algebra and applications | year=2003 | location=London | publisher=Springer-Verlag | isbn=1-85233-667-6 | zbl=1006.00001 }}

Category:Algebraic structures