semisimple ring

A ring $R$ is (left) semisimple if it satisfies one of the following equivalent statements:

1. All left $R$ -modules are semisimple.
2. All finitely-generated left $R$ -modules are semisimple.
3. All cyclic left $R$ -modules are semisimple.
4. The left regular $R$ -module $_RR$ is semisimple.
5. All short exact sequences of left $R$ -modules split.

The last equivalent condition offers another homological characterization of a semisimple ring:

• A ring $R$ is (left) semisimple iff all of its left modules are projective.

A more ring-theorectic characterization of a (left) semisimple ring is:

• A ring is left semisimple iff it is semiprimitive and left artinian.

In some literature, a (left) semisimple ring is defined to be a ring that is semiprimitive without necessarily being (left) artinian. Such a ring (semiprimitive) is called Jacobson semisimple, or J-semisimple, to remind us of the fact that its Jacobson radical is (0).

Relating to von Neumann regular rings, one has:

• A ring is left semisimple iff it is von Neumann regular and left noetherian.

The famous Wedderburn-Artin Theorem states that a (left) semisimple ring is isomorphic to a finite direct product of matrix rings over division rings.

The theorem implies that a left semisimplicity is synonymous with right semisimplicity, so that it is safe to drop the word left or right when referring to semisimple rings.