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

1.
All left $R$modules are semisimple.

2.
All finitelygenerated (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 ringtheorectic 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 Jsemisimple, 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 WedderburnArtin 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  20130322 14:19:05 
Last modified on  20130322 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 