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Чистая математика и смежные философские проблемы.

In 1945, N. Jacobson has introduced the concept of radical of a ring A (which is known as "Jacobson radical", and is denoted J=J(A)). Later the notion of (Jacobson) radical of a left (or right) A-module M, J(M), has been defined as the intersection of all submodules N of M such that M/N is simple. Thus one may consider the left radical Jl=J(AA) of A considered as a left module over itself and the right radical Jr=J(AA) of A considered as a right module over itself. These are bilateral ideals of A, and are contained in J(A). If A has identity, one has J=Jl=Jr, but this equality is not valid in general. Dually, it is possible to define left socle Sl and the right socle Sr of A. We shall establish relations between J, Jl, Jr, Sl and Sr, and for artinian algebras we shall obtain expressions for Jl(A), Jr(A), Sl(A) and Sr(A). In particular, if A is a finite dimensional algebra over a ! field we display Jl(A) (and Jr(A)) in a matrix representation.

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