This short post discusses a cute statement which shows the power of a certain subset of axioms of a ring, including prominently the distributive law.
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.