Answer to Question #23253 in Abstract Algebra for Hym@n B@ss

Question

Let p ⊂ R be a prime ideal, A be a left ideal and B be a right ideal. Does AB ⊆ p imply that A ⊆ p or B ⊆ p?

Answer 1

The answer is “no.” For instance,let R be any prime ring with an idempotent e not equal 0, 1.(We can take R = Mn(Z) with n ≥ 2.) Then, p = 0 is a primeideal in R. However, for A = Re nonzero and B = (1 − e)Ralso nonzero, we have AB = Re(1 − e)R = 0.