Exploring the Structure of Positive Rational Numbers as a Free Module

This blog serves as a reimagined take on a previous discussion, extending its scope while not encompassing everything from before.

Before delving into this post, it is recommended to read Math Essays: Coprime-Orthogonal, Module and p-adic valuation.

We explore the monoid $(\mathbb{N},\times )$$(\mathbb{N}, \times)$ and its Grothendieck Group $({\mathbb{Q}}^{+},\times )$$(\mathbb{Q}^+, \times)$, with the goal of demonstrating that $({\mathbb{Q}}^{+},\times )$$(\mathbb{Q}^+, \times)$ forms a free $\mathbb{Z}-$$\mathbb{Z}-$module.

According to the fundamental theorem of arithmetic, any positive rational number $q\in {\mathbb{Q}}^{+}$$q \in \mathbb{Q}^+$ can be expressed uniquely as a product of prime powers:

$$q=2^{v_2(q)}{3}^{v_3(q)}{5}^{v_5(q)}\dots$$

This leads us to the mapping:

$$q\mapsto (v_2(q),v_3(q),v_5(q),\dots )$$

Proposition: The group of positive rational numbers under multiplication, $(\mathbb{Q},\times )$$(\mathbb{Q}, \times)$, is isomorphic to ${\mathbb{Z}}^{\oplus P}$$\mathbb{Z}^{\oplus P}$. Here, $P\cong \mathrm{Spec}\mathbb{Z}$$P \cong \mathrm{Spec}\,\mathbb{Z}$ represents the set of prime numbers.

Proof: A free module, by definition, is ${\mathbb{Z}}^{\oplus P}:=\{f:P\to \mathbb{Z}|f(p)=0\text{ for all but finitely many }p\in P\}$$\mathbb{Z}^{\oplus P} := \{f: P \to \mathbb{Z} | f(p) = 0 \text{ for all but finitely many } p \in P\}$.

Each function $f\in {\mathbb{Z}}^{\oplus P}$$f \in \mathbb{Z}^{\oplus P}$ can be uniquely represented as:

$$f=\sum _{n\in P}{c}_p{\delta }_p$$

where $c_n$$c_n$ are integers with only a finite number of them being nonzero, and ${\delta }_p$$\delta_p$ is defined by:

$$\begin{array}{c}{\delta }_p:=\{\begin{array}{l}1\text{ if }x=p\\ 0\text{ if }x\ne p\end{array}\end{array}$$

It becomes evident that ${\delta }_p$$\delta_p$ corresponds to $v_p$$v_p$.

The isomorphic relationship is established through the mapping:

$$\sum _{p\in P}{c}_p{v}_p⟼\prod _{p\in P}{p}^{c_p}$$