Classify abelian groups $A$ which are irreducible $End(A)$-modules

Question

Classify abelian groups $A$ which are irreducible $End(A)$-modules.

I think i did it for finite abelian group $A$ .

A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. What about infinite case i think prüfer groups , $\mathbb{Z}$ and $\mathbb{Q}$ is not irreducible but i can't give any general statement for infinite abelian group.Please give hint .

Answer 1

Hint. For any integer $n\geq 2$, there are $\mathrm{End}(A)$-stable subgroups $$A_n=\lbrace\,a\in A\mid na=0\,\rbrace$$ and $$nA=\lbrace\,a\in A\mid \exists\alpha\in A,\,a=n\alpha\,\rbrace\,.$$ Edit. I don't have a complete answer for when $A$ is infinite, but, assuming the axiom of choice, the answer is: $A$ is irreducible iff it has a vector space structure (i.e. $A$ is the underlying group of some vector space). There are two cases for any $\mathrm{End}(A)$-irreducible abelian group $A\neq 0$.

1. Either, for all prime numbers $p$, $A_p=0$, $pA=A$, and $A$ is a $\Bbb Q$-vector space, and $\mathrm{End}_{Ab}(A)=\mathrm{End}_{\Bbb Q}(A)$. If $A$ is finite dimensional, then $A$ will be $\mathrm{End}_{Ab}(A)$ irreducible without further assumptions, indeed, if we fix $a\in A\setminus\lbrace0\rbrace$ and any $a'\in A$, there exists many $\Bbb Q$-linear maps sending $a$ to $a'$. The same holds true if $A$ has a $\Bbb Q$-basis, for instance if we assume the axiom of choice.
2. Otherwise, there exists a single prime number $p$ with $A_p=A$ and $pA=0$, with for all other primes $q$, $A_q=0$ and $qA=A$. $A$ is then naturally a $\Bbb Z/p\Bbb Z$-vector space, and, again, $\mathrm{End}_{Ab}(A)=\mathrm{End}_{\Bbb Z/p\Bbb Z}(A)$. If $A$ is finite, then $A\simeq(\Bbb Z/p\Bbb Z)^n$ is irreducible by the same vector space argument as above, while if $A$ is infinite, the answer will likely depend on some form of the axiom of choice. If $A$ is known to have a basis, it certainly is true that it will be irreducible over $\mathrm{End}_{Ab}(A)$.