Jacobson Radical Theory

Abstract

Historically, the notion of the radical was a direct outgrowth of the notion of semisimplicity. It may be somewhat surprising, however, to remark that the radical was studied first in the context of nonassociative rings (namely, finite-dimensional Lie algebras) rather than associative rings. In the work of E. Cartan, the radical of a finite-dimensional Lie algebra A (say over ℂ) is defined to be the maximal solvable ideal of A: it is obtained as the sum of all the solvable ideals inA. The Lie algebra A is semisimple iff its radical is zero, i.e., iff it has no nonzero solvable ideals. Cartan characterized the semisimplicity of a Lie algebra in terms of the nondegeneracy of its Killing form, and showed that any semisimple Lie algebra is a finite direct sum of simple Lie algebras. Moreover, he classified the finite-dimensional simple Lie algebras (over ℂ). Therefore, the structure theory of finite-dimensional semisimple Lie algebras is completely determined.KeywordsDirect SummandCommutative RingLeft IdealGroup RingDivision RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.