Suppose that

is a set with addition and multiplication defined that satisfy all of the axioms of a ring except possibly for

(commutativity of addition). Prove that

, and hence that

is a ring.

Now, we want to show that the group

is abelian. Since, not every group is abelian, we will have to use the properties of multiplication. The only axiom that links addition to multiplication is the axiom of distributivity so if my reasoning is correct the proof should use that axiom. However, I am a little stuck on this and would really appreciate any hints.