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I found they sometimes are both written in $A/N$ pronounced $(A \bmod N)$

A remainder ring? a quotient group?

what's the difference?

Are there any conventional problems in notations?

HyperGroups
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    Look up "quotient group" and "quotient ring" to see for yourself. – anon Jun 29 '13 at 03:14
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    @annon ok, good to know the word quotient ring. I like asking in SE and one of my purpose is forcing myself writing in English. – HyperGroups Jun 29 '13 at 03:26
  • Yes. It is the quotient by that subgroup given the structure of the ring. – Potato Jun 29 '13 at 03:30
  • @HyperGroups, it is very good to want to write in English, but usually you should not ask questions which are answered by standard sources like Wikipedia! – Mariano Suárez-Álvarez Jun 29 '13 at 03:32
  • @MarianoSuárez-Alvarez ok, learned that, sometimes I'm daunting at the large material in Wikipedia, of course they are so good. If I've read some materials like Wikipedia, and cannot understand clearly, is that good to ask here for some explanation? – HyperGroups Jun 29 '13 at 03:39
  • Yes, indeed, HyperGroups: that's a fine approach...and when you do post here, after checking out Wikipedia, mention that you read Wiki...but don't understand ....(fill in the blank). – amWhy Jun 29 '13 at 04:07
  • @HyperGroups: I don't know if this is still relevant to you, but you may consider using chat for this purpose. – Eric Stucky Feb 19 '16 at 04:35

2 Answers2

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$A/N$ isn't necessarily a quotient ring: just as with groups, the notation $\,A/N\,$ denotes the set of cosets of $\,N\,$ in $\,A.\,$ When $N$ is normal in group $A$, we have that $\,A/N\,$ is a quotient group (aka "factor group"). Similarly, when $N = I$ is an ideal in ring $A$, we have that $\,A/N\,$ a quotient ring.:

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor [quotient] groups of group theory and the quotient spaces of linear algebra. One starts with a ring $R$ and a two-sided ideal $I$ in $R$, and constructs a new ring, the quotient ring $R/I$, essentially by requiring that all elements of $I$ be zero in $R$. Intuitively, the quotient ring $R/I$ is a "simplified version" of $R$ where the elements of $I$ are "ignored".

See also this earlier post: What is a quotient ring and cosets?. You may find the question and the accepted answer to be very helpful, too.

Community
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amWhy
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In both cases, $A/N$ means the set of cosets of $N$ in $A$, endowed with binary operation(s) defined by their action on coset representatives.

Gerry Myerson
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