Intereting Posts

Eigenvalues of an $n\times n$ symmetric matrix
Differential topology book
Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.
Proving $\pi(\frac1A+\frac1B+\frac1C)\ge(\sin\frac A2+\sin\frac B2+\sin\frac C2)(\frac 1{\sin\frac A2}+\frac 1{\sin\frac B2}+\frac 1{\sin\frac C2})$
$R/M$ is a division ring
Inverse of an invertible triangular matrix (either upper or lower) is triangular of the same kind
Cover $\mathbb{R}^3$ with skew lines
Point Group of a pattern
Stuck trying to prove an inequality
Is this convergent or divergent: $\sum _{n=1}^{\infty }\:\frac{2n^2}{5n^2+2n+1}$?
Prove $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$
Find all natural numbers $n$ such that $21n^2-20$ is a perfect square.
Why is the Jordan Curve Theorem not “obvious”?
Show that $\lim_{n\to\infty}n+n^2 \log\left(\frac{n}{n+1}\right)= 1/2$
Does proper map $f$ take discrete sets to discrete sets?

I’m trying to solve this for a problem and I need to know if what I have done is right:

Let $B$ be a torsion free abelian group. Then we consider the set $$A=\{(b,n):b\in B,n\in \mathbb Z,n\neq 0\}$$ and define

$$(b,n)\sim (a,m) \text{ iff } bm=an.$$

This yields an equivalence relation, now you can define addition of classes by $(b,n)+(a,m)=(am+bn,nm)$. Then $(A,+)$ is a torsion free abelian group and $B$ can be embedded into $(A,+)$, but also $(A,+)$ is divisible, so $(A,+)$ can be embedded in a direct sum of copies of $\mathbb Q$.

Thank you for your time.

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Here is an answer using tensor products, inspired in egreg’s comment above.

By corollary 4.27 in this expository article by Keith Conrad, the map $B\to B\otimes_\mathbb{Z} \mathbb{Q}$, $b\mapsto b\otimes 1$ is injective, since $B$ is torsion free. Now, the abelian group $B\otimes_\mathbb{Z} \mathbb{Q}$ has a $\mathbb{Q}$-vector space structure (it is the $\mathbb{Q}$-extension of scalars of $B$), hence it is isomorphic to a direct sum of copies of $\mathbb{Q}$.

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