Help on notation: $\mathbb{Z}/n\mathbb{Z}$ vs. $\mathbb{Z}_n$ I have difficulties understanding the difference between the following two notations:


*

*$\mathbb{Z}/n\mathbb{Z}$ (which denotes a quotient ring) and

*$\mathbb{Z}_n$.


Are they equivalent?
PS1: The same applies to the multiplicative counterparts:


*

*$(\mathbb{Z}/n\mathbb{Z})^*$

*$\mathbb{Z}_n^*$.


PS2: It can be proven that $\mathbb{Z}/n\mathbb{Z}$ is a ﬁeld if and only if $n$ is prime. Assuming $n$ is prime, could you compare $\mathbb{Z}/n\mathbb{Z}$ with $\text{GF}(n)$?
 A: If $n$ is a prime number, then $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$ are isomorphic (in fact I would simply define $GF(n)=\mathbb{Z}/n\mathbb{Z}$ when $n$ is a prime number).  
However, if $n$ is some power of a prime number, say $n=p^k$ for $k\geq 2$, then $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$ are not the same. 
A: The notations are equivalent if the author has been careful enough to tell you that by $Z_n$ she means "the integers modulo $n$." If she has not been careful than you have to study the context to decide whether the author means the integers modulo $n$ or something else. 
By the way, $Z/nZ$ is not just a quotient group, it's a quotient $\it ring$ (if you haven't studied rings and ideals yet, you have something to look forward to!). 
A: It depends on the textbook/paper author, but often $\mathbf{Z}/n\mathbf{Z}$ and $\mathbf{Z}_n$ mean the same thing. 
A word of caution, however: using the notation $\mathbf{Z}_n$ to mean $\mathbf{Z}/n\mathbf{Z}$ can cause confusion, because $\mathbf{Z}_p$ is also used to denote the p-adic integers. Thus, many mathematicians (especially number theorists) reserve the shorter notation for p-adics and use the long notation for the finite cyclic groups.
Edit: Just now saw your second question. The answer is that, indeed,  $\mathbf{Z}/p\mathbf{Z} = GF(p)$, where $p$ is prime.
A: To avoid confusion that mentioned in Jeff's answer, some contemporary textbooks (like Rotman's Advanced Modern Algebra) use $\mathbb I_n$ instead of $\mathbb Z_n$. The symbol $\mathbb I$ is the first letter of integer.
A: I very like the notation $\mathbb Z_{/n}$ to denote the ring of integers modulo $n$. This avoids ambiguities.
