Why is ring addition commutative? What is the motivation behind axiomatically forcing the underpinning group of a ring to be abelian? Noncommutative rings are vastly more complex than commutative ones, so I am assuming that allowing the additive operation to be noncommuting would just make matters worse. Is there something deeper here, or is this a restriction for the sake of convenience and simplicity? 

$\left(1+1\right)\left(a+b\right)=1\left(a+b\right)+1\left(a+b\right)=a+b+a+b$
$\left(1+1\right)\left(a+b\right)=\left(1+1\right)a+\left(1+1\right)b=a+a+b+b$
So distributivity (on both sides) demands that $a+b=b+a$
