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

Question #23253

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?

Expert's answer

The answer is “no.” For instance,let

*R*be any prime ring with an idempotent*e*not equal 0*,*1.(We can take*R*= M*n*(Z) with*n ≥*2.) Then, p = 0 is a primeideal in*R*. However, for A =*Re*nonzero and B = (1*− e*)*R*also nonzero, we have AB =*Re*(1*− e*)*R*= 0.
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