semisimple ring

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

  1. 1.

    All left R-modules are semisimple.

  2. 2.

    All finitely-generated ( left R-modules are semisimple.

  3. 3.

    All cyclic left R-modules are semisimple.

  4. 4.

    The left regularPlanetmathPlanetmath R-module RR is semisimple.

  5. 5.

    All short exact sequencesMathworldPlanetmathPlanetmath of left R-modules split (

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 (

A more ring-theorectic characterization of a (left) semisimple ringMathworldPlanetmath 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 radicalMathworldPlanetmath is (0).

Relating to von Neumann regular ringsMathworldPlanetmath, one has:

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

The famous Wedderburn-Artin Theorem that a (left) semisimple ring is isomorphicPlanetmathPlanetmathPlanetmath to a finite direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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