Generic matrix ring

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

Definition

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We denote by   a generic matrix ring of size n with variables  . It is characterized by the universal property: given a commutative ring R and n-by-n matrices   over R, any mapping   extends to the ring homomorphism (called evaluation)  .

Explicitly, given a field k, it is the subalgebra   of the matrix ring   generated by n-by-n matrices  , where   are matrix entries and commute by definition. For example, if m = 1 then   is a polynomial ring in one variable.

For example, a central polynomial is an element of the ring   that will map to a central element under an evaluation. (In fact, it is in the invariant ring   since it is central and invariant.[1])

By definition,   is a quotient of the free ring   with   by the ideal consisting of all p that vanish identically on all n-by-n matrices over k.

Geometric perspective

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The universal property means that any ring homomorphism from   to a matrix ring factors through  . This has a following geometric meaning. In algebraic geometry, the polynomial ring   is the coordinate ring of the affine space  , and to give a point of   is to give a ring homomorphism (evaluation)   (either by Hilbert's Nullstellensatz or by the scheme theory). The free ring   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

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For simplicity, assume k is algebraically closed. Let A be an algebra over k and let   denote the set of all maximal ideals   in A such that  . If A is commutative, then   is the maximal spectrum of A and   is empty for any  .

References

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  1. ^ Artin 1999, Proposition V.15.2.
  • 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.