semisimple ring
A ring is (left) semisimple if it one of the following statements:
- 1.
All left -modules are semisimple.
- 2.
All finitely-generated (http://planetmath.org/FinitelyGeneratedRModule) left -modules are semisimple.
- 3.
All cyclic left -modules are semisimple.
- 4.
The left regular
-module is semisimple.
- 5.
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 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.