In this post, we prove the following well-known lemma:

**Lemma.** *Let satisfy all axioms of a ring, except possibly the commutativity . Then is abelian.*

That is, additive commutativity of a ring is implied by the other axioms.

*Proof.* By distributivity, we have , so multiplication by is a homomorphism. By our previous post, this implies is abelian.