semisimple ring

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

1. All left $R$-modules are semisimple.

2. All finitely-generated (http://planetmath.org/FinitelyGeneratedRModule) left $R$-modules are semisimple.

3. All cyclic left $R$-modules are semisimple.

4. The left regular $R$-module ${}_{R}R$ is semisimple.

5. All short exact sequences of left $R$-modules split (http://planetmath.org/SplitShortExactSequence).

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

• A ring $R$ is (left) semisimple iff all of its left modules are projective (http://planetmath.org/ProjectiveModule).

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 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.

Title	semisimple ring
Canonical name	SemisimpleRing
Date of creation	2013-03-22 14:19:05
Last modified on	2013-03-22 14:19:05
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	11
Author	CWoo (3771)
Entry type	Definition
Classification	msc 16D60
Related topic	SemiprimitiveRing
Defines	semisimple