TITLE: Endomorphism ring which is not the endomorphism ring of an abelian group.
QUESTION [5 upvotes]: I have to find an example of a ring that can't be the endomorphism ring of an abelian group.

I am studying Fuchs, "Abelian groups", but I haven't seen such an example.
Does anyone have some references?

REPLY [2 votes]: Well, Krylov certainly includes a brief discussion of these in his book (Amazon link). But these rings aren't so exotic (necessarily). For example, the rings $\pmatrix{ \mathbb{Z} & \mathbb{Q} \\ 0 & \mathbb{Q}}$ and $\mathbb{Q} \times \mathbb{Q}$ are such rings.