Do silly-rings exist?

Question

A ring can be defined as a near-ring satisfying two-sided distributivity, whose underlying additive group is Abelian. Negating this second stipulation, we obtain the following definition. A silly-ring is a near-ring satisfying two-sided distributivity, whose underlying additive group is non-Abelian.

Do silly-rings exist?

Answer 1

Yes, they exist. Let $(R,+)$ be any non-Abelian group. Define $xy=0$ for all $x,y\in R$. (Ask a silly question, . . .)

Answer 2

The comments show that if near-rings are assumed to have unity, then silly-rings do not exist. Here's a slightly more general proposition, following along the lines of bof's comment.

Proposition. Consider a structure $(R,+,\cdot)$ such that

1. Two sided distributivity holds.
2. $+$ is associative
3. Both cancellation laws hold.

Then for all $x,y \in R$, we have that if $x$ factors as $ab$ and $y$ factors as $cd$, then $x+y=y+x$.

Proof. Distribute $(a+c)(b+d)$ both ways, and cancel.