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单词 SemisimpleRing
释义

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 (http://planetmath.org/FinitelyGeneratedRModule) 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 (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 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.

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更新时间:2025/5/4 16:54:37