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?

  • $\begingroup$ $R+I$ would usually be interpreted as $\{r+i\mid r\in R, i\in I\}$, so that is likely to cause confusion. $\endgroup$ – rschwieb Jan 21 '14 at 14:49

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).

  • $\begingroup$ I find the "order" argument for the notation especially compelling. I wonder if the slash notation has its roots in equivalence relations. @martin Rubey It's common practice to write $X/\sim$ for the classes of an equivalence relation $\sim$ on $X$, and since there's no operation involved the question about using $+$ becomes kind of moot. $\endgroup$ – rschwieb Jan 21 '14 at 14:53
  • $\begingroup$ @rschwieb I suppose one could also backwards engineer a possible root of the notation that way as well, with each $I$ defining an equivalence relation on $R$ by $x\sim y$ iff $x-y\in I$, so that $R/I=R/{\sim}$. $\endgroup$ – mdp Jan 21 '14 at 15:05
  • $\begingroup$ That's what my question was: "does the use of the notation in equivalence relations predate the special-case use here?" $\endgroup$ – rschwieb Jan 21 '14 at 15:36
  • $\begingroup$ @rschwieb Ha, sorry, missed that. I read your whole comment once, and then when I wanted to remind myself what you said about equivalence relations I started again from the @. Oops. $\endgroup$ – mdp Jan 21 '14 at 15:39

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.

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    $\begingroup$ I don't mean to necromance here but this post does occasionally get upvotes still so I feel I should make myself clearer. My main point is that arranging objects into equivalence classes is very similar to the process of taking quotients. In particular if one wants to figure out how many dollars one has if one has 100 quarters, one divides by four. In doing so one "bundles" or "packages" the set of 100 quarters into 25 packs of 4. I feel that this process is very similar to packaging a ring into packs of $I$, which one does in taking a quotient by an ideal. This is how I think about it anyway. $\endgroup$ – Sempliner Jun 24 '16 at 5:46

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