nLab semisimple ring



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

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


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

commutative ringreduced ringintegral domain
local ringreduced local ringlocal integral domain
Artinian ringsemisimple ringfield
Weil ringfieldfield


  • 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.

Last revised on October 3, 2023 at 18:12:23. See the history of this page for a list of all contributions to it.