By O.U. Schmidt
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Such that an E anRa n) . Specializing the above result, show that the following are also equivalent: (A) R is reduced (no nonzero nilpotents), and K -dim R = O. (B) R is von Neumann regular.
J» Solution. By symmetry, it is sufficient to prove the above "Double Annihilator Property" for I . 7), and let f = 1 - e. We claim that annr(I) = f R. Indeed, since I · f R = RefR = 0, we have f R ~ annr(I) . Conversely, if a E ann, (I) , then ea = 0 so a = a - ea E f R . This proves annr(I) = f R, and hence similarly anne (annr(I» = anne(JR) = Re = I . Comment. An artinian ring satisfying the Double Annihilator Properties in this Exercise is known as a quasi-Probenius ring. The conclusion of the Exercise is therefore that any semisimple ring is a quasi-Probenius ring .
Specializing the above result, show that the following are also equivalent: (A) R is reduced (no nonzero nilpotents), and K -dim R = O. (B) R is von Neumann regular.
Abstract Theory of Groups by O.U. Schmidt