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?
 A: 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). 
A: 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.
