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)_ \mid 1 \le l \le m,\ 1 \le i, j \le n generated by ''n''-by-''n'' matrices $X_1, \dots, X_m$, where $(X_l)_$ 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)_ 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 , \dots, t_m/math> 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 , \dots, t_m\to k (either by the Hilbert 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 $\operatorname_n(A)$ denote the set of all maximal ideals $\mathfrak$ in ''A'' such that $A/\mathfrak \approx M_n(k)$. If ''A'' is commutative, then $\operatorname_1(A)$ is the maximal spectrum of ''A'' and $\operatorname_n(A)$ is empty for any $n > 1$.

References

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* {{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

Algebraic structures