(no subject)

[leblon]

Is it true that an algebra whose Jacobson radical vanishes is semi-simple (in the usual sense)? Or at least is it true for finite-dimensional algebras?

Comments

[posic.livejournal.com]

Then it is certainly true for finite-dimensional algebras and certainly not true for infinite-dimensional ones.

In the former case, one usually proves the stronger assertion that over a finite-dimensional algebra without two-sided nilpotent ideals, the category of left modules is semi-simple.

For a counterexample in the latter case, take the algebra of polynomials in one variable (or in several variables, or the universal enveloping algebra of a Lie algebra, or the algebra of differential operators on an affine space, etc.)

[buddha239.livejournal.com]

+100!:)

[leblon.livejournal.com]

Спасибо! Я нашел доказательство в книжке Lam, A First Course on Noncommutative Rings.