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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