Simple Artinian rings are matrix rings over division rings

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

Simple Artinian rings are matrix rings over division rings

A central structure theorem in ring theory is that “finite-length” (Artinian) simple rings are precisely matrix rings over division rings. This is the simplest nontrivial case of the Artin–Wedderburn theorem .

Theorem

Let $R$ be a ring (not necessarily commutative). Assume:

$R$ is simple (it has no nonzero proper two-sided ideals), and
$R$ is Artinian (equivalently, it is Artinian as a left or right module over itself).

Then there exist an integer $n\ge 1$ and a division ring $D$ such that

R \cong M_n(D)

as rings, where $M_n(D)$ denotes the ring of $n\times n$ matrices over $D$ .

A useful refinement is that $D$ may be taken to be the endomorphism ring of a simple right $R$ -module (a division ring by Schur’s lemma), and $n$ corresponds to the multiplicity with which that simple module appears in a decomposition of $R$ as a module over itself. The general semisimple case (finite products of such matrix rings) is encoded in the semisimple Artinian product decomposition .

In the commutative setting, this theorem collapses strongly: if $R$ is commutative and simple Artinian, then necessarily $n=1$ and $D$ is commutative, so $R$ is a field .

Examples

Matrix rings over a field.
For any field $k$ and any $n\ge 1$ , the ring $M_n(k)$ is simple Artinian. Here $D=k$ .
Division rings (the case $n=1$ ).
Any division ring $D$ is simple Artinian, and the theorem recovers it as $M_1(D)\cong D$ . For instance, the real quaternions $\mathbb H$ form a (noncommutative) division ring, hence are simple Artinian.
Why “simple” matters.
The ring $k\times k$ is Artinian and semisimple, but not simple: it has nontrivial two-sided ideals $k\times 0$ and $0\times k$ . Accordingly, it is not a single matrix ring, but it fits the product form described by semisimple Artinian product decomposition .

This theorem is frequently used alongside Jacobson radical facts such as the Jacobson radical annihilates simple modules , since in the semisimple Artinian setting the Jacobson radical is zero.