Coprime-Orthogonal, Module and p-adic valuation

Coprime-Orthogonal, Module and p-adic valuation.

We know that according to the fundamental theory of arithmetic,

we can represent the number of ${\mathbb{N}}^{+}$$\mathbb N^+$ , $m=p_1^{i_1}{p}_2^{i_2}{p}_3^{i_3}...$$m=p_1^{i_1}p_2^{i_2}p_3^{i_3}...$as $(i_1,i_2,...,i_k,...$$(i_1,i_2,...,i_k,...$

it look like a vector and then, $gcd(a,b)=1$$\gcd(a,b)=1$ can be view as $a\mathrm{\perp }b$$a\bot b$

And recently, I have been learning module theory

And I observe that if we let $i\in \mathbb{Z}$$i\in\mathbb Z$, not only in $\mathbb{N}$$\mathbb N$, We can get a $\mathbb{Z}-\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}$$\mathbb Z-\rm Module$

The Module is a representation of $({\mathbb{Q}}^{+},\times )$$(\mathbb Q^+,\times)$

$(i_i,i_2,...,i_k,...\pm (j_1,j_2,...,j_k=(i_i\pm j_1,i_2\pm j_2,...,i_k\pm j_k,...$$(i_i,i_2,...,i_k,...\pm(j_1,j_2,...,j_k=(i_i\pm j_1,i_2\pm j_2,...,i_k\pm j_k,...$

is a representation of $\times ,÷$$\times,\div$

$\lambda (i_1,i_2,...,i_k,...=(\lambda i_1,\lambda i_2,...,\lambda i_k,...$$\lambda(i_1,i_2,...,i_k,...=(\lambda i_1,\lambda i_2,...,\lambda i_k,...$

is the power, $a^{\lambda }$$a^\lambda$

And in this case, we can view the p-adic valuation $v_p(Q)$$v_p(Q)$ as a projection!

That is a really natural way to think about it.