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What the meaning of the slash "/" in expression like this: $\mathbb{Z}/p\mathbb{Z}$ ? I know that it is called "a quotient ring", but quotient reminds division. It's an operator somewhat related to division?

I'm an engineer trying to fully understand the abstract algebra behind error correction codes. May you leave a suggestion of a book for non-mathematicians that explain this topic in an easy manner?

Bernard
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Berk7871
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  • (It's not very related to division of numbers in a way that would be easily recognizable to everyone who can execute division... but has some figurative comparisons...) – paul garrett Aug 14 '17 at 22:52
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    For further usage, the standard unit circle in radians, where angles are identified when they differ by an integral multiple of $2\pi$, could be written as $\mathbf R/2\pi\mathbf Z$. – KCd Aug 14 '17 at 22:54
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    if you still want a book, I've thought Bob Ash's book on abstract algebra covers way more than you will need and is a nice resource. – mdave16 Aug 14 '17 at 23:17
  • If $G$ is a group with $n$ elements, and $H$ is a normal subgroup with $m$ elements, then the so-called "quotient group" $G/H$ has $n/m$ elements (actual division). – GEdgar Aug 15 '17 at 00:12

4 Answers4

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If $R$ is a ring and $J$ a two-sided ideal of $R$, the quotient ring $R/J$ consists of the equivalence classes $x + J$ for $x \in R$, where $x \sim y$ if $x - y \in J$. This is a ring with operations $(x+J) + (y+J) = (x+y)+J$ and $(x+J)(y+J) = xy + J$.

In the case of $\mathbb Z / p\mathbb Z$, $p\mathbb Z$ consists of the multiples of $p$ and the equivalence relation is congruence mod $p$. Thus $\mathbb Z/p\mathbb Z$ consists of the congruence classes mod $p$.

Robert Israel
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Risking self-praise, I might recommend my own course notes (formerly a book published by a traditional publisher) "Coding Notes" at http://www.math.umn.edu/~garrett/coding/CodingNotes.pdf There are also the rather telegraphic overheads for a course I taught many times on that subject using those notes/book, at http://www.math.umn.edu/~garrett/coding/

This course (and the book/notes) was meant to be intelligible to people who'd not studied any abstract algebra before, and, in particular, to engineering and computer science people, in addition to math majors in the relatively early part of their undergrad education.

So, in particular, these notes are very down-to-earth, and talk in a way precisely meant to be intelligible to engineering and computer science people... who may have a "different dialect" in mathematics.

So, no, not abstract, yet mathematically accurate, and aimed at coding-theory issues. (Though not really high-end, and certainly no longer up-to-date.)

paul garrett
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  • is 13 years really enough for a set of notes to be no longer up-to-date? (I can imagine a few things have improved, but as a whole the area should be the same, no?) – mdave16 Aug 14 '17 at 23:25
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    @mdave16 The basics are the same, but the currently-optimal codes (like LDPC, low-density parity-check codes) I didn't treat at the time. Things like that. Nothing's "wrong" in my notes/book, but some (now-we-know) good things are omitted. So "not up to date" is an important disclaimer, I fear. – paul garrett Aug 14 '17 at 23:30
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In $\Bbb Z$ ,we define the equivalence relation $R$ by $x \;R\; y \iff x-y$ is a multiple of $p $.

$\Bbb Z/p \Bbb Z $ is the set of equivalence classes.

$$\Bbb Z/3\Bbb Z=\{\overline {0},\overline {1},\overline {2}\}. $$

$$\overline {3}=\overline {0}. $$

hamam_Abdallah
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The generic notation for the quotient of a set $X$ by an equivalence relation $\cal R$ is: $\; X/\cal R$.

As the equivalence relation here can be defined as $\;x\cal R y\iff x-y\in p\mathbf Z$, the ideal $p\mathbf Z$ is taken as the name of the relation and used in the notation of the quotient set.

Bernard
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