For those who are unfamilar with nearrings, here is a definition. Note that there are left-nearrings (where only the left distribution property is assumed), and right-nearrings (where only the right distribution property is assumed) as well. I only consider two-sided nearings.
Definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both left and right distributive properties of the multiplication $\cdot$ over the addition $+$, namely, $$x\cdot(y+z)=(x\cdot y)+(x\cdot z) \text{ and }(x+y)\cdot z=(x\cdot z)+(y\cdot z)$$ for $x,y,z\in N$. Of course, every additive group $(G,+)$ with identity $0_G$ can be made a two-sided nearring with the trivial multplication $g\cdot h:=0_G$ for all $g,h\in G$. Such a nearring is called a trivial two-sided nearring.
My question is about nontrivial two-sided nearrings which are not rings (i.e., the addition $+$ is not commutative). I know one which has $128$ elements: $N:=(\mathbb{Z}/8\mathbb{Z})\times (2\mathbb{Z}/8\mathbb{Z})\times (2\mathbb{Z}/8\mathbb{Z})$, where $$\left(a_1,b_1,c_1\right)+\left(a_2,b_2,c_2\right):=\left(a_1+a_2+c_1b_2,b_1+b_2,c_1+c_2\right)$$ and $\left(a_1,b_1,c_1\right)\cdot\left(a_2,b_2,c_2\right)$ is given by $$\left(a_1b_2+a_2b_1+a_1c_2+a_2c_1+b_1b_2+b_2c_1+c_1c_2,b_1b_2+b_1c_2+b_2c_1+c_1c_2,0\right)\,,$$ for all $a_1,a_2\in\mathbb{Z}/8\mathbb{Z}$, and $b_1,b_2,c_1,c_2\in2\mathbb{Z}/8\mathbb{Z}$.
Old Question. Can you find a nontrivial two-sided nearring which is not a ring with the minimum number of elements ? This question has been answered here.
The example seen in Eran's answer in the link above is a two-sided nearring which is not a ring with the smallest number of elements. While the addition of this example is noncommutative, the multiplication is commutative. Therefore, I am offering a bounty price for anybody who answers the question below.
Bounty Question. What is a two-sided nearring with the minimum number of elements which is not a ring and whose multiplication is also noncommutative? Please also prove that your example has the smallest number of elements amongst all two-sided nearing whose multiplication and addition are noncommutative.
A Remark (which may or may not be helpful). If $(N,+,\cdot)$ is a two-sided nearing, then the subnearring $N^{\cdot 2}$ of $N$ generated by elements of the form $a\cdot b$, where $a,b\in N$, is a ring. That is, $N^{\cdot 2}$ consists of all integer combinations of $a\cdot b$, where $a,b\in N$. To show this, let $a,b,c,d\in N$. Then, we have $$(a+b)\cdot (c+d)= a\cdot(c+d)+b\cdot(c+d)=(a\cdot c+a\cdot d)+(b\cdot c+b\cdot d)$$ and $$(a+b)\cdot(c+d)=(a+b)\cdot c+(a+b)\cdot d=(a\cdot c+b\cdot c)+(a\cdot d+b\cdot d)\,,$$ whence $$a\cdot c+a\cdot d+b\cdot c+b\cdot d=a\cdot c+b\cdot c+a\cdot d+b\cdot d\,,$$ making $$a\cdot d+b\cdot c=b\cdot c+a\cdot d\,.$$ In particular, if $N$ is a two-sided nearring which is not a ring, then $N$ cannot have a multiplicative identity (otherwise, $N^{\cdot 2}=N$, making $N$ a ring).