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Answer on Algebra Question 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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