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I am confused about the following statement my lecture slides make. The context is they have just finished proving the chinese remainder theorem, and the easy chinese remainder theorem:

Finally, we note that $x$ can be represented by $(a, b)$ where $x \equiv a(\bmod n)$ and $x \equiv b \pmod m$, and given $x$ and $y$ with representations $\left(x_a, x_b\right)$ and $\left(y_a, y_b\right)$, that $x+y$ has the representation $\left(x_a+y_a, x_b+y_b\right)$. A similar observation works for multiplication, which implies that arithmetic $\pmod {mn}$ acts in the same manner as arithmetic over the pairs of numbers viewed in modulo $m$ and $n$ respectively. Thus, in addition to a bijection between these sets, this mapping between $\{0, \ldots, m n-1\}$ and $\{0, \ldots, n-1\} \times\{0, \ldots, m-1\}$ are considered isomorphic under these operations as well. This relationship extends in the same manner to a case where there are more than two relatively prime moduli. I have 3 questions here:

  1. what does it mean to say that arithmetic $\pmod {mn}$ is the same as arithmetic over pairs of numbers viewed in modulo $m$ and $n$ respectively? One the one hand we are dealing with pairs of numbers, on the other hand numbers not in pairs, so I am having trouble making the connection.
  2. What does "this mapping between $\{0, \ldots, m n-1\}$ and $\{0, \ldots, n-1\} \times\{0, \ldots, m-1\}$ are considered isomorphic under these operations as well" mean? What does this have to do with the pairwise operations being the same as $\pmod {mn}$?
  3. Am I correct to say that the generalization to more than 2 coprime numbers, such as $a,b,c,d$, would be that doing arithmetic in a tuple of four elements for each of these moduli is the same as doing arithmetic in $\pmod {abcd}$?
Princess Mia
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  • and 2. are about the CRT. The map gives a ring isomorphism $\Bbb Z_{mn}\cong \Bbb Z_m\times \Bbb Z_n$ for coprime $m,n$. So the arithmetic of a pair of rings corresponds to the arithmetic of a singe ring - because they are isomorphic.
    • – Dietrich Burde Aug 25 '23 at 18:11
  • @dietrichBurde I am not familiar with rings and they haven't been taught in this course; how could I translate what you have said into non-ring terminology? – Princess Mia Aug 25 '23 at 18:14
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    Your text is already the translation. You can take an example like $m=2$, $n=3$ an writing down the two and three elements (numbers). The answers to your questions will involve rings with the CRT. If you don't want to know about the rings $\Bbb Z/n\Bbb Z$, then you have to take the "vague" description "the arithmetic acts in the same manner as" ... – Dietrich Burde Aug 25 '23 at 18:18
  • Shmuel, All this requires that $m,n$ be coprime. Your last paragraph, point 3, suggests that you were told that at some point. – Will Jagy Aug 25 '23 at 18:54
  • Without knowledge of ring theory, you can view it as meaning that $\Bbb Z_{mn}$ and the set of pairs $,\Bbb Z_m\times \Bbb Z_n, $ (with pointwise operations) $\bf\text{have the same addition and multiplication tables},$(up to order). Pair arithmetic in $,\Bbb Z_m\times \Bbb Z_m,$ is done pointwise $,(x,y)+(x',y') = (x+x',y+y'),$ and $,(x,y)(x',y')=(xx',yy').,$ Since the operation tables are essentially the same it follows by induction that all arithmetic is essentially the same in both structures. – Bill Dubuque Aug 25 '23 at 20:40
  • For an explicit example see here where I show the isomorphism for an addition table in $\Bbb Z_7$ with that for a shifted addition. You may find it instructive to construct such isomorphic addition and multiplication tables for a CRT isomorphism (for some small $,m,n)$. – Bill Dubuque Aug 25 '23 at 20:41
  • @BillDubuque I have read your linked post, but I am still confused about the sense in which pairwise operations have the same addition and multiplication tables as $\mathbb{Z}_{mn}$ (the table you drew seems to be for single numbers rather than pairs). Are they the same in the sense that adding x and y where these are both congruent to $a \pmod m$ and $b \pmod n $ is in pairwise $(x_a + y_a, x_b + y_b)$, which is considered the same thing as $x+y \pmod {mn}$? In what way is this pairwise result the same thing as just adding them in $\pmod {mn}$? – Princess Mia Aug 25 '23 at 23:33
  • @Shmuel There are some examples here, e.g. see the final displayed equations. – Bill Dubuque Aug 26 '23 at 02:14