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?
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.)
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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.)
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