We know a ring R is semisimple ring iff every module over R is semisimple，a ring R is von-Neumann regular ring iff every module over R is flat，What about the Jacobson semisimple ring？
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An alternative phrasing to Bugs's answer is as follows: A ring $R$ is Jacobson semisimple if and only if the only element which annihilates every simple $R$-module is the zero element. In my experience (which is largely in the commutative case, or even finite type algebras over a field; in this latter case Jacobson semisimple coincides with reduced), this point of view on Jacobson semisimple rings is one that comes up frequently in arguments. |
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Mighty Wikipedia to the rescue!! Seriously I dont think you can say anything better than a semisimple faithful module exists... What is the cause of your curiosity? Having said that, I can think of a cute reformulation that makes it clear that the property is Morita-invariant: for any projective $P$, the hom-space $Hom (P, \oplus_i S_i)$ is a faithful $End (P)$-module where the sum is taken over non-isomorphic simples in the category. |
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