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

Question

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

Answer 1

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.