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Simple Artinian rings are matrix rings over division rings

A simple Artinian ring is isomorphic to a full matrix ring over a division ring.

Let RR be a nonzero ring, not necessarily commutative. If RR is simple, meaning that it has no nonzero proper two-sided ideals, and left Artinian, meaning that every descending chain of left ideals stabilizes, then there are an integer n≥1n\ge 1 and a DD such that

R≅Mn(D)R\cong M_n(D)

as rings. Conversely, every full matrix ring Mn(D)M_n(D) over a division ring is simple Artinian. This is the simple case of the .

Refinements

A useful refinement is that DD may be taken, up to passing to the opposite ring according to module conventions, from the endomorphism ring of a simple RR-module. The general semisimple case is the into such matrix rings.

In the commutative setting, necessarily n=1n=1 and DD is commutative, so RR is a .

Examples
  1. Matrix rings over a field. For any field kk and any n≥1n\ge 1, the ring Mn(k)M_n(k) is simple Artinian.
  1. Division rings (the case n=1n=1). Any division ring DD is simple Artinian, and M1(D)≅DM_1(D)\cong D. For instance, the real quaternions H\mathbb H form a noncommutative division ring.
  1. Why “simple” matters. The ring k×kk\times k is Artinian and semisimple, but not simple: it has the nontrivial ideals k×0k\times0 and 0×k0\times k. It therefore appears as a product rather than as one matrix-ring factor.