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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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    Yes, they are. But $\mathbb{Z}_p$ often denoted $p$-adic integers which is not the same as $\mathbb{Z}/p\mathbb{Z}$ at all. Some author(e.g. Rotman) also uses $\mathbb{I}_n$ denote $\mathbb{Z}/n\mathbb{Z}$2011-06-07
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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

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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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    For some reason, some authors and teachers think that they will scare first year undergraduates by using the quotient group/ring notation. That is the notation one should exclusively though, for the reason you describe.2011-06-07
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    I agree! Another notation which I don't often see, but which I sort of favor, is $\mathbf{C}_n$ for the cyclic group of order n. It has the same brevity as $\mathbf{Z}_n$ without the confusion. Unless $\mathbf{C}_n$ is used for something else, too...?2011-06-07
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    $C_n$ or $Z_n$ (for German zyklisch) is fine and is often used, but $\mathbb{C}_p$ is used in number theory for the completion of the algebraic closure of the field of $p$-adic numbers, so shouldn't be used in this context.2011-06-07
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    I disagree that the $\textbf{Z}/n\textbf{Z}$ notation should be used exclusively. When dealing with direct sums of several cyclic groups (e.g. in homology computations), the $\textbf{Z}_n$ notation is much more readable. I would also feel strange using $C_n$ for an additive cyclic group.2011-06-07
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    @Jim I am objecting to bold face. See my second comment.2011-06-07
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    Good points, all. As long as authors are explicit at the offset about what notation they are using, I guess it needn't cause any confusion. As always, context is key.2011-06-07
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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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    @Zev using the undefined notation $GF(n)$ that didn't feature anywhere in the question and not answering the actual question is likely to exacerbate the OP's confusion instead of clearing it up.2011-06-07
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    @Alex: The OP amended their question to also ask for a comparison of $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$.2011-06-07
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    @Zev: sorry Zev, I hadn't see the edit. +1 then :-)2011-06-07
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    Thanks. As you pointed, the wording "it is said" is not good. I changed it to "it can be proven."2011-06-07
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    @Sadeq: I've removed the corresponding part of my answer, now that it is changed.2011-06-07
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    By the way, $GF(n)$ is sometimes denoted $\mathbb{F}_n$.2011-06-07
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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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    This was mentioned in a comment to the original question. Five years ago, ten minutes after the question was posted.2016-03-24
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    @tomasz :) I only read the question and the answers!2016-03-24