transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
A semisimple ring $R$ is one obeying any of the following equivalent conditions:
$R$ is an Artinian ring with vanishing Jacobson radical.
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.
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.
commutative ring | reduced ring | integral domain |
---|---|---|
local ring | reduced local ring | local integral domain |
Artinian ring | semisimple ring | field |
Weil ring | field | field |
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.