Is it true that all modules over $\mathbb{Q}$ are of form $\mathbb{Q}^n$ where $n\geq 0$

Question

Is it true that all modules over $\mathbb{Q}$ are of form $\mathbb{Q}^n$ where $n\geq 0$

I feel like this is true because $\mathbb{Q}$ is a principal ideal domain

Answer 1

Since $\Bbb Q$ is a field, a module over it is also a vector space. Vector spaces always permit a basis (unless we are denied the Axiom of Choice). Hence every finitely generated module over $\Bbb Q$ is isomorphic to some $\Bbb Q^n$ with $n\in\Bbb N_0$. However, there are of course also infinte-dimensional vector spaces, i.e., for any set $A$, the set $\Bbb Q^A$ of maps $A\to \Bbb Q$ forms a vector space, which is finte -dimensional only if $A$ is finite.

Answer 2

All unitary modules over $\mathbb{Q}$ are $\mathbb{Q}$-vector spaces. There are non-unitary modules over $\mathbb{Q}$. In fact, every module over $\mathbb{Q}$ can be written as $V\oplus N$, where $V$ is a $\mathbb{Q}$-vector space and $N$ is a trivial module (i.e., an abelian group such that $q\cdot x=0$ for all $q\in\mathbb{Q}$ and $x\in N$). However, your definition of modules may already assume that they must be unitary.