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Just getting started with modular arithmetic.

Are the elements of a modular ring simply the set of all the numbers from $1$ to $p-1$? in this case $p-1 = 6$ ?

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  • $\begingroup$ $0$ is also included. $\endgroup$ – paw88789 Sep 30 at 15:50
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A "mod $p$ ring" is constituted by numbers from $0$ to $p-1$, so:

$$\mathbb{Z}_p:=\{0, 1, ..., p-2, p-1\}$$

This rings have the property that "$p=0$". Addition and multiplication work just as they do on the integers, but adding that particular property, thus, in your example, for $p=7$, you can say:

$$3+5=8=7+1=0+1 = 1$$ $$5 \cdot9=45=6 \cdot 7 + 3 = 6 \cdot 0 + 3 = 3$$

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No, strictly speaking, the elements of that ring are not the usual counting numbers from $0$ to $6$.

Often we use those numbers to name the elements of the ring, but redefine addition and multiplication, so that, for example, "$3 + 5 = 1$", which we write as $$ 3 + 5 \equiv 1 \pmod{7}. $$

I think that's what you need to know to when

getting started with modular arithmetic.

What the elements of that ring actually are - how you define that ring - depends on the level of abstraction you start from.

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  • $\begingroup$ The first and last paragraphs contradict each other. I'd fix it by correcting the first paragraph (in fact in many elementary expositions the answer is "yes"). $\endgroup$ – Bill Dubuque Sep 30 at 16:42
  • $\begingroup$ @BillDubuque I don't think so, since calling those elements "numbers" implies that they behave the way numbers behave under addition and multiplication. I agree that elementary expositions correctly elide that, which is what my second paragraph is meant to express. The last paragraph covers the definition as elements of a quotient ring, or as arithmetic progressions (in this example). $\endgroup$ – Ethan Bolker Sep 30 at 17:05
  • $\begingroup$ You notion of "number" seems to be much more restricted than that of most number theorists or algebraists. The first paragraph is very misleading as it stands (and it contradicts many elementary expositions). $\endgroup$ – Bill Dubuque Sep 30 at 17:33
  • $\begingroup$ @BillDubuque But the OP is not a number theorist or algebraist. I think in the question "number" means an ordinary counting number. I think we have to agree to disagree. I doubt anyone will be seriously confused. But I will add a bit ... $\endgroup$ – Ethan Bolker Sep 30 at 17:37
  • $\begingroup$ So, to you, are $2\in\Bbb C,\ 2\in\Bbb Z[x]$ "numbers"? Ditto for $2+i$ and $2+x\in \Bbb Z[x]$ and the matrix $[2]$? $\endgroup$ – Bill Dubuque Sep 30 at 17:40
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Short answer to you question:

Yes, if you include $0$ as well. The elements are $\{0,1,2,....,p-1\}$.

More obtuse answer is .... Wellllll, what exactly is a "number" anyway? Is the $2$ here the same $2$ well I taught my son that $2 + 47=49$. Is it the same $2$ that I'm gossiping about when I spread rumors like $2^k-1 =\sum\limits_{k=0}^{k-1} 2^k$ or "$\sqrt 2$ is irrational"? And is it the same $2$ as when I said "Two people came to my party"?

Normally, we dismiss this as pedantic abstraction but in modular arithmetic it has practical applications.

We often define there is a modular relationship between two numbers $x,y$ (and these are regular old numbers we are used when we learned to count in kindergarten or when you learned about real numbers in high school) that $x$ and $y$ are related if $x-y= k*p$ for some integer $k$. This is an equivalence relationship and this relationship is maintain over basic arithimetic: That is if $x-y = k*p$ for some integer $k$ and $w-z= m*p$ for some integer $m$ then $(x+w)-(y+z)=(k+m)p$; so if $x$ is related to $y$ and $w$ is related to $z$ then $x+y,x-y, x\cdot y$ are related to $w+z, w-x, w\cdot z$.

In this way we think of a integer $k$ in the ring not as the number $k$, but as the set of all possible integers that are related to $k$

So $\overline k = \{...., k-2p, k-p, k, k+p, k+2p, k+3p,.....\}$

And $\overline k + \overline j := \overline{k+j}=\{.....,(k+j)-2p, (k+j)-p,(k+j), (k+j)+2p, ....\}$.

This is a subtle point that is eludes many students the first time they see it. It eluded me.

Although in practice it amounts to the same thing.

......

But yes the elments of the modular ring mod 7 are $\{0,1,2,3,4,5,6\}$

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