semisimple ring (original) (raw)

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

1. 1. All left R-modules are semisimple.
2. 1. All finitely-generated (http://planetmath.org/FinitelyGeneratedRModule) left R-modules are semisimple.
3. 1. All cyclic left R-modules are semisimple.
4. 1. The left regular R-module RR is semisimple.
5. 1. All short exact sequences of left R-modules split (http://planetmath.org/SplitShortExactSequence).

The last condition offers another homological characterization of a semisimple ring:

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  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 ring is:

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  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).

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  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.