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.