A ring is (left) semisimple if it one of the following statements:
All left -modules are semisimple.
All finitely-generated (http://planetmath.org/FinitelyGeneratedRModule) left -modules are semisimple.
All cyclic left -modules are semisimple.
The left regular -module is semisimple.
All short exact sequences of left -modules split (http://planetmath.org/SplitShortExactSequence).
The last condition offers another homological characterization of a semisimple ring:
A ring 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 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.
|Date of creation||2013-03-22 14:19:05|
|Last modified on||2013-03-22 14:19:05|
|Last modified by||CWoo (3771)|