The Question
Is it possible to characterise the abelian groups $G$ whose endomorphism rings are Dedekind-finite? Or, at least, are there conditions which are necessary/sufficient for this to occur?
(A ring $R$ is said to be Dedekind-finite if $ab=1\implies ba=1$.)
Some Motivation
In algebraic structures such as groups and rings, the inverse of a homomomorphism $f:A\to B$, if it exists, is also a homomorphism. This leads to the rather convenient fact that a bijective homomorphism is automatically an isomorphism. Unfortunately, the situation with injective homomorphisms is not so nice: if $f:A\to B$ is an injective homomorphism, then it is not necessarily the case that a left inverse of $f$ is also a homomorphism; likewise, a right inverse of a surjective homomorphism is not necessarily a homomorphism.
These considerations led me to the following question. Suppose $G$ is an abelian group, and $R$ is its endomorphism ring. If $R$ is not Dedekind-finite, then there exist $f,g\in R$ such that $f\circ g=\DeclareMathOperator{\id}{id}\id$ but $g\circ f\neq\id$. The map $g$ has the following properties: it is an injective, but not surjective, homomorphism, and it has a left inverse $f$ which is also a homomorphism. This situation can occur for some rings, e.g. in the case that $G=\mathbb R^{\mathbb N}$ with pointwise addition, but not for others, e.g. $G=\mathbb Z$ with the usual addition. In the latter case, every injective endomorphism which is not surjective, such as $f(n)=2n$, does not have a left inverse which is also an endomorphism.
The following is one elementary set of equivalents:
A very closely related condition is: an $R$ module $M$ is called Hopfian if every onto endomorphism of $M$ is an isomorphism. (An Abelian Hopfian group is just a Hopfian $\mathbb Z$ module) It turns out this is sufficient for $End(M_R)$ to be Dedekind finite but it is not equivalent in general.