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?

We do so because in general what we are doing is arranging the object $R$ into equivalence classes (in such a way that the set of equivalence classes has a structure analogous to that of $R$), in a manner very similar to what happens when one takes one integer modulo another (in fact this can be reconceptualized as the quotient of the ring $\mathbb Z$ by one of its ideals $n\mathbb Z$). This is a very general operation done in many objects in mathematics, and it is almost always referred to as a quotient. Further $R + I$ in most contexts refers to something like the set of all $r + i$, where $i \in R, i \in I$, whereas $R/I$ refers to the set of equivalence classes $\hat{r}$, where $r, s \in \hat{r}$ if $r - s \in I$. This can be understood as simply $r + I$ but it is not the same as the set of ALL $r + I$ as above, because (with notation as before) $r + I$ and $s + I$ are the same thing. There is more structure there than just taking sums of things.
