# Answer to Question #23464 in Algebra for john.george.milnor

Question #23464

Show that a ring R is semiprime iff, for any two ideals A,B in R, AB = 0 implies that A ∩ B = 0.

Expert's answer

First assume

*R*is semiprime,and let AB = 0. For C : = A*∩*B, we have C^2 = CC*⊆**AB = 0, so C = 0. Conversely, assume the implication property about A**,*B,and consider any ideal C with C^2 = 0. Then CC = 0 implies 0 = C*∩*C =C, so*R*is semiprime.
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