I don't know much abstract algebra or ring theory. I have come across a ring, and it would be really great if somebody could help me to understand what else in mathematics it relates to, or how to visualize what is going on in this ring (especially for multiplication).
The underlying set is $\mathbb R \times \mathbb R,$ so the elements are pairs of real numbers. Addition of $(a,b)$ and $(c,d)$ is defined as $(a,b) \oplus (c,d) = (a + c, b + d).$ Multiplication of $(a,b)$ and $(c,d)$ is defined as $(a,b) \odot (c,d) = (a \times c, (a \times d) + (b \times c)).$
I noticed that if we write $(a,b)$ as $b/a$ then $\oplus$ and $\odot$ look like Farey addition, and standard fraction addition, respectively. Although I suspect somebody can point out much deeper relationships between this ring and other ideas from mathematics. I'm not even sure what the ring is called. It is mentioned in the Kock-Lawvere Axiom.