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The Jacobson radical of a finite-dimensional algebra is the intersection of its maximal left ideals
Definition
Let be a finite-dimensional algebra over a field. Its Jacobson radical is
Because the left regular module is finitely generated, maximal proper left ideals exist by Under Choice, every finitely generated nonzero module has a maximal proper submodule whenever . On finite-dimensional algebras this radical agrees with the usual right-sided definition, so the notation is unambiguous here.
Depends on
Used by
- For a finite group and a field of characteristic p, the group algebra is local exactly when the group is a p-group Theorem
- For a finite-dimensional algebra, the Jacobson radical is nilpotent and the quotient by it is semisimple Theorem
- For a finite-dimensional algebra, the module radical is exactly the action of the Jacobson radical Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)