Rings with noncommutative addition I was wondering if "rings" with noncommutative addition are studied at all? Of course, if a ring $R$ has a $1$, then for all $a, b\in R$, $a+a+b+b=(1+1)a+(1+1)b=(1+1)(a+b)=(a+b)+(a+b)=a+b+a+b$, from which it follows from cancellation that $a+b=b+a$. Thus, rings with $1$ automatically must have commutative addition. 

So, are there interesting, necessarily non-unital, "rings" with noncommutative addition? Of course, you can take any non-abelian group and make such a "ring" by having all multiplications give the additive identity. Are there more interesting examples?
Vectors on a sphere? Their addition is essentially multiplication of rotations, which is non-commutative.
Vectors on a sphere? Their addition is essentially multiplication of rotations, which is non-commutative.