Today, I learned about a very exciting statement when reading a thread in my favourite math forum.

Assume that you have an algebraic structure with two binary operations and , where is distributive over . Further assume that (a) there exists a neutral element with respect to , and that (b) the operation is associative, and that (c) the operation is cancelable, i.e. for we have and . Note that (b) and (c) are satisfied for example if forms a group.

These rather weak requirements already imply that is commutative: if , then

Using the cancellation property, implies .

Using the aforementioned special case that forms a group, we obtain:

Assume that the algebraic structure satisfies that is a group and is distributive over and has a neutral element. Then is already an abelian group.

Therefore, if we loosen up the definition of a unitary ring by dropping the requirement that addition is commutative, the other axioms already force the commutativity of addition. Therefore, to get something more general than unitary rings (even if the multiplication is not associative or commutative), one has to make sure that does not imply , for example by asking for an addition not having the cancelation property.

## Comments.

Hi Felix, I just came across your math blog incidentally and found (again) something interesting. :)

There is actually the notion of "near-ring", which is - besides semirings - another generalization of rings. Here, only a one-sided distributive law is postulated, and hence the addition might actually be noncommutative.

The endomorphisms of a group form such a near-ring, which is probably the most important example.

Thanks for the info, Jens. I didn't knew about "near-rings" yet. Also the example - endomorphisms of a group - is very helpful. I knew that they don't form a ring (unless the group is abelian, or maybe in some other random cases), but didn't knew the structure they form had a name.