Let $I =\{ n \mathbb{Z} : n \in \mathbb{Z} \}$ be the inverse system of nonzero ideals of $\mathbb{Z}$. Define an $I$-cauchy sequence in $\mathbb{Q}$ to be a sequence $\{ a_n \}_{n \in \mathbb{N}_{\geq 1}} $ such that, for each ideal $\mathfrak{a}$ of $\mathbb{Z}$, there is $n \in \mathbb{N}_{\geq 1}$ such that $a_m - a_l \in \mathfrak{a}$ for $m, l \geq n$. We put an equivalence relation on the set $C$ of $I$-cauchy sequences of $\mathbb{Q}$, where $\{ a_n \}_{n \in \mathbb{N}_{\geq 1}} \sim \{ b_n \}_{n \in \mathbb{N}_{\geq 1}}$ when for each ideal $\mathfrak{a} \in I$, there is $n \in \mathbb{N}_{\geq 1}$ such that $a_m - b_m \in \mathfrak{a}$ for each $m \geq n$. Then define $\hat{\mathbb{Q}}$ to be $C / \sim$, with the apparant ring structure:
$$ 0 = [ \{ 0\}_{n \in \mathbb{N}_{\geq 1}} ] $$
$$ 1 = [ \{ 1 \}_{n \in \mathbb{N}_{\geq 1}} ] $$
$$ [ \{ a_n\}_{n \in \mathbb{N}_{\geq 1}} ] + [ \{ b_n \}_{n \in \mathbb{N}_{\geq 1}} ] = [ \{ a_n+ b_n \}_{n \in \mathbb{N}_{\geq 1}} ] $$
$$ [ \{ a_n\}_{n \in \mathbb{N}_{\geq 1}} ] \cdot [ \{ b_n \}_{n \in \mathbb{N}_{\geq 1}} ] = [ \{ a_n b_n \}_{n \in \mathbb{N}_{\geq 1}} ] $$
Question: Is $\hat{\mathbb{Q}}$ the ring of finite adeles?
One might suspect that this ring resembles the ring $\mathbb{Q} \otimes_{\mathbb{Z}} \hat{\mathbb{Z}}$. Firstly, a similar theorem holds for $\hat{\mathbb{Z}}$; the elements of $\hat{\mathbb{Z}} = \text{lim} \mathbb{Z} / n \mathbb{Z}$ can be expressed as cauchy sequences in a similar fashion.
Also, there is for each prime a map $\hat{\mathbb{Q}} \rightarrow \mathbb{Q}_p$, which gives a map into the product $\hat{\mathbb{Q}} \rightarrow \prod_{p \in \mathbb{N}_{\geq 1}, p \text{ prime }} \mathbb{Q}_p$.
Sure $$\Bbb{Z}_p= \varprojlim \Bbb{Z}/(p^k)$$ is the set of limits of sequences of integers that converge modulo $p^k$ for all $p^k$,
$$\Bbb{Q}_p=\Bbb{Q}+\Bbb{Z}_p$$ is the set of limits of sequences of rationals $(x+a_j)_j$ such that $(a_j)$ is a sequence of integers that converges in $\Bbb{Z}_p$,
$$\hat{\Bbb{Z}}= \varprojlim \Bbb{Z}/(n) =\prod_p \Bbb{Z}_p$$ is the set of limits of sequences of integers that converge modulo $n$ for all $n$, a ring with the termwise addition and multiplication of sequences
$$\Bbb{A_{Q,fin}}=\Bbb{Q} +\hat{\Bbb{Z}}= \prod_p' \Bbb{Q}_p$$ is what we obtain when we consider the limits of sequences of rationals $(x+a_j)_j$ such that $(a_j)_j$ is a sequence of integers that converges $\bmod n$ for all $n$.
It is again a ring with the pointwise addition/multiplication because given $l$ the denominator of $x$ there is an integer $b$ such that $a_j-b\to 0\bmod l$.