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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 $\text{GF}(n)$?

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    Usually I use the notation $GF(p)$ or $\mathbb F_p$ when I'm explicitly interested in the field structure. For the additive group structure I prefer $\mathbb Z/p\mathbb Z$ or $C_p$.2011-06-07

4 Answers 4

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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.

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    while $\mathbb{Z}_{(p)}$ denotes localized at $(p)$...2011-06-07
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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.

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    @Alex: No problem, and thanks for the +1 :)2011-06-07
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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!).

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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.

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    @tomasz :) I only read the question and the answers!2016-03-24