Are Mod of Rings are the same to Quotient Groups? 
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?
 A: $A/N$ 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.
A: 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. 
