For $R = \mathbb Z_2 \times \mathbb Z_2$ we can define a product by the following table:
\begin{matrix} \cdot & a & b & c & d \\ a & a & a & a & a \\ b & a & a & d & d \\ c & a & a & c & c \\ d & a & a & b & b \end{matrix}
where $a = (0,0), b = (0,1), c= (1,0), d= (1,1). $ We can verify that $R$ is indeed a ring. I've checked that this product is not commutative: $bc=d\neq a = cb,$ is not associative: $(db)c = ac = a \neq b = dd = d(bc)$ and clearly does not have a unit element.
My question is: Can I generalize this result? Is there a result concerning those kind of products (non-associative, non-commutative and non-unital) for a ring $R$ with more than 4 elements?
Observation: I am asking this because my professor posed us a question to define such a product on $\mathbb Z_2\times \mathbb Z_2$. At the end of the exercise, he asked: Can you generalize for $R$ with $\geq4 $ elements?