Let $(\mathbb{Z}_l-\text{mod})\otimes \mathbb{Q}_l$ be the category whose objects are $\mathbb{Z}_l$ modules and morphism groups are tensored with $\mathbb{Q}_l$, my understanding of this is like just kill all the torsion modules and think two morphisms as the same if they differ by a map to a torsion submodule.

My question is,

Is $(\mathbb{Z}_l-\text{mod})\otimes \mathbb{Q}_l\cong \mathbb{Q}_l-\text{Vect}$ ?

The only thing that makes me worry is whether this is true without any finiteness condition. E.g. I'm convinced that

$ \text{(Finite rank abelian groups)} \otimes_\mathbb{Z}\mathbb{Q} \cong \text{Finite-dimensional-} \mathbb{Q}\text{-vector-spaces}$.

As every $\mathbb{Q}-$ matrix is a $\mathbb{Z}-$matrix times a rational number. But I'm not sure if it is true without the finite-rank condition.

The question make sense for any domain $R$, I use $\mathbb{Z}_l$ instead of $R$ just because I thought about this when reading \'etale cohomology. (Maybe dim$R$ = 1 makes a difference?)