TITLE: What is the characteristic of the module over Jacobson semisimple ring？
QUESTION [3 upvotes]: 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？

REPLY [3 votes]: 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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