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Definition

A semisimple ring $R$ is one obeying any of the following equivalent conditions:

• $R$ is an Artinian ring with vanishing Jacobson radical.

• $R$ is a finite product of simple Artinian rings.

• As a left module over itself, $R$ is a semisimple object in the category of left $R$-modules.

• As a right module over itself, $R$ is a semisimple object in the category of right $R$-modules.

• The category of finitely generated left $R$-modules is a semisimple category.

• The category of finitely generated right $R$-modules is a semisimple category.

• Every left $R$-module is projective.

• Every right $R$-module is projective.

A commutative ring is semisimple if and only if it is both Artinian and a reduced ring.

 Examples

By the Wedderburn-Artin theorem, every semisimple ring is a finite product of matrix algebras over division rings.

As a special case, every semisimple commutative ring is a finite product of fields. In particular, every semisimple integral domain is a field. Similarly, every semisimple local ring is a field.

Beware: not every simple ring is a semisimple ring. A simple ring is semisimple if and only if it is Artinian. For example the Weyl algebra over any field is a simple ring that is not Artinian and not semisimple: intuitively speaking, it is “too infinite” to be Artinian, which is a kind of finiteness condition.

See also

 References

• Wikipedia, Semisimple ring.

• Frank W. Anderson and Kent R. Fuller, Rings and Categories of Modules, Springer Graduate Texts in Mathematics, Vol. 13, 2012. Chapter 4, Section 13: Semisimple rings.