Notation for quotient ring I'm wondering where the notation for the quotient of a ring by an ideal comes from.  I.e., why do we write $R/I$ to denote a ring structure on the set $\{r+I: r\in I\}$, wouldn't $R+I$ be more natural?

When $A,B\subseteq R$, with $R$ a ring, it is common to write
$$A+B=\{a+b:a\in A,b\in B\}$$
This is particularly useful when $A$ and $B$ are ideals, in which case $A+B$ is also an ideal. So $R+I$ already has an interpretation (although it would just be $R$).
On the other hand, if $R$ is finite, then the number of elements of $R/I$ is $|R|/|I|$. I believe this is the origin of the corresponding notation for groups (and then it makes sense to use the same notation for quotients of all types of algebraic object). 
