In a lecture, our professor gave an example for a ring. He took it out of another source and mentioned that he does not know the motivation for the chosen operation.
Of course, it's likely that somebody just invented an arbitrary operation satisfying ring axioms. I'd still like to try my luck whether anyone here can decipher the operation and give any kind of motivation for that example.
On $\mathbb{R}^3$ define the operations $+$ and $\cdot$ by $$ \begin{aligned} (a_1, a_2, a_3) + (b_1,b_2,b_3) &= (a_1+b_1,a_2+b_2,a_3+b_3) \\ (a_1, a_2, a_3) \cdot (b_1, b_2, b_3) &= (a_1 \cdot b_1, a_2 \cdot b_2, a_1 \cdot b_3 + a_3 \cdot b_2). \end{aligned} $$ (The $+$ and $\cdot$ operations on the right side are the usual addition and multiplication from $\mathbb{R}$.) With those operations, one can confirm that $\left(\mathbb{R}^3, +, \cdot \right)$ is a ring.