How should I think about this ring? 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.
 A: This is a common construction in ring theory called a trivial extension or split-null extension in the literature.
It can be thought of as the quotient $\mathbb R[x]/(x^2)$, where $(a,b)$ corresponds to $a+bx+(x^2)$ in the quotient.  In this form the construction is also named dual numbers, but that monicker is not applied more generally, really.
Here’s a second way: Given an $R,R$ bimodule $M$, one can think of the set of matrices of the form $\begin{bmatrix}a&m\\ 0&a\end{bmatrix}$ where $a\in R$ and $m\in M$ as a ring using matrix addition and multiplication.  When you abbreviate that matrix to $(a,m)$, and $M=R$ is commutative, it boils down to the quotient above.
I mentioned this second way because it sort of shows a connection to a more general construction called a triangular matrix ring. Given an $R, S$ bimodule $M$, you can form a ring out of $\begin{bmatrix}R&M \\ 
 0&S\end{bmatrix}$ with the obvious operations.  The second way above is a subring of that.
Most generally if you have an $R,S$ bimodule $M$ and an $S,R$ bimodule $N$, you have a ring $\begin{bmatrix}R&M\\ N&S\end{bmatrix}$ which is called a Morita context ring.
A: This is just $\Bbb R[\varepsilon]/(\varepsilon^2)$ where $(a,b)$ corresponds to $[a+b\varepsilon]$.
In algebraic geometry, it is related to tangent spaces.
