Help on notation: $\mathbb{Z}/n\mathbb{Z}$ vs. $\mathbb{Z}_n$

Question

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 $\text{GF}(n)$?

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 $\mathbb{Z}/n\mathbb{Z}$ — wxu, Jun 07 '11 at 06:20
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 $C_p$. — Giacomo d'Antonio, Jun 07 '11 at 08:02

Jeff · Accepted Answer · 2011-06-07T06:19:36.610

12

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.

edited Jun 07 '11 at 06:19

answered Jun 07 '11 at 06:13

Jeff

2,594

2

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. – Alex B. Jun 07 '11 at 06:17
3

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...? – Jeff Jun 07 '11 at 06:20
1

$C_n$ 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. – Alex B. Jun 07 '11 at 06:22
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. – Jim Belk Jun 07 '11 at 06:23
@Jim I am objecting to bold face. See my second comment. – Alex B. Jun 07 '11 at 06:23
1

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. – Jeff Jun 07 '11 at 06:25
1

while $\mathbb{Z}_{(p)}$ denotes localized at $(p)$... – wxu Jun 07 '11 at 06:32

Zev Chonoles · Answer 2 · 2011-06-07T06:27:00.033

7

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.

edited Jun 07 '11 at 06:27

answered Jun 07 '11 at 06:16

Zev Chonoles

129,973

@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. – Alex B. Jun 07 '11 at 06:19
@Alex: The OP amended their question to also ask for a comparison of $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$. – Zev Chonoles Jun 07 '11 at 06:21
@Zev: sorry Zev, I hadn't see the edit. +1 then :-) – Alex B. Jun 07 '11 at 06:24
Thanks. As you pointed, the wording "it is said" is not good. I changed it to "it can be proven." – Sadeq Dousti Jun 07 '11 at 06:27
@Sadeq: I've removed the corresponding part of my answer, now that it is changed. – Zev Chonoles Jun 07 '11 at 06:27
By the way, $GF(n)$ is sometimes denoted $\mathbb{F}_n$. – Jim Belk Jun 07 '11 at 06:29
@Alex: No problem, and thanks for the +1 :) – Zev Chonoles Jun 07 '11 at 06:29

score 4 · Answer 3 · answered Jun 07 '11 at 06:25

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!).

score 1 · Answer 4 · answered Mar 24 '16 at 07:46

1

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.

answered Mar 24 '16 at 07:46

This was mentioned in a comment to the original question. Five years ago, ten minutes after the question was posted. – tomasz Mar 24 '16 at 08:09
1

@tomasz :) I only read the question and the answers! – Mar 24 '16 at 08:11

score 1 · Answer 5 · answered Oct 25 '21 at 16:22

1

I very like the notation $\mathbb Z_{/n}$ to denote the ring of integers modulo $n$. This avoids ambiguities.

answered Oct 25 '21 at 16:22

MikeTeX

2,078

Help on notation: $\mathbb{Z}/n\mathbb{Z}$ vs. $\mathbb{Z}_n$

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