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.

Hilarious. I didn’t even know this.

Exercise: Find a counterexample for the same statement but for (R, +, •, 0) (i.e., without unit). (It’s actually not hard to find a minimal one, as I know you like to do!)