Distributive nearring

Solution 1:

Consider the dihedral group $D_4=\langle a,b :a^4, b^2, (ab)^2\rangle$. Define $x+y=xy$ and $x\cdot y=[x,y]$. Claim: $(D_4,+,\cdot)$ is a distributive near-ring.

Recall commutator identities $[xy,z]=[x,z]^y[y,z]$ and $[z,xy]=[z,y][z,x]^y$. Note that the commutator subgroup coincides with the center. So we have $[xy,z]=[x,z][y,z]$ and $[z,xy]=[z,y][z,x]=[z,x][z,y]$. This translates to $(x+y)\cdot z=x\cdot z+y\cdot z$ and $z\cdot(x+y)=z\cdot x+z\cdot y$.

More generally, this works for any group of nilpotence class 2. For more info see this paper: Heatherly, H. E. Distributive near-rings. Quart. J. Math. Oxford Ser. (2) 24 (1973), 63–70.