I was going through some of my notes in abstract algebra (group and ring theory) and came across this $$R/S$$ where $R$ is a ring, $S$ is an additive subgroup of $R$ and $R/S$ is the additive quotient group. Now I understand the theory of cosets and how they work for normal subgroups etc, but the notation confuses me. I am unable to visualize the quotient group (same for ideals) because I am not able to understand the operation. Could you please clarify how exactly does $/$ work?

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    $\begingroup$ The quotient is done basically only in the additive group. So you take cosets as you would if $R$ were just the additive group (forgetting about the ring structure.) It turns out that if $S$ is an ideal, you can define a multiplication to turn the quotient into a ring. $\endgroup$ – Siddharth Venkatesh May 5 '14 at 12:44
  • $\begingroup$ Ah ok :) thank you :) $\endgroup$ – Artemisia May 5 '14 at 12:45
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    $\begingroup$ To expand upon Siddharth's comment, and using the canonical map $R\to R/S$, elements of $S$ are sent to the additive identity in $R/S$. In any ring, the additive identity is $0$. So if $R/S$ is to be a ring, then the additive identity needs to behave like $0$ (i.e. $r\cdot 0 = 0\cdot r = 0$ for any $r$). It turns out that this condition in $R/S$ exactly corresponds to the condition that $S$ is an ideal in $R$. $\endgroup$ – Arthur May 5 '14 at 12:52

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