Let $R$ be a ring satisfying $(xy)^3=xy$ for all $x,\, y\in R$. Then $R$ is commutative.

Question

Let $R$ be a ring satisfying $(xy)^3=xy$ for all $x,\, y\in R$. Then $R$ is commutative. Any suggestion how to prove it?

I don't think this is an exact duplicate. The ring might have no unit element. — azimut, Aug 01 '13 at 11:36
@azimut possible duplicate, I agree with Chris. See e.g. the answer of Ehsan M. Kermani, where he/she writes Re-posting it from here. Note that R is not necessarily unital. — vesszabo, Aug 01 '13 at 11:59
Since we are not in the habit of mentally adding hypotheses to trivialize questions: not a duplicate. — rschwieb, Aug 01 '13 at 12:46

score 3 · Answer 1 · answered Aug 01 '13 at 12:42

3

An answer is the proof of Theorem 15 from

answered Aug 01 '13 at 12:42

Boris Novikov

1 Answers1