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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?

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    Related: http://math.stackexchange.com/questions/2540562016-08-18

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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.

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    @stef: When I first studied about these rings, these were a natural result of a theorem or lemma (or something). I don't know how you've approached the subject, nor anything about Fuchs. So let me say a third - $\mathbb{F}_p \times \mathbb{F}_p$ is another such group. This is easier to see, as if it were End($G$), then $\mathbb{F}_p$ acts on $G$ and thus it would have to be an $\mathbb{F}_p$ vector space - which it's not. Similarly, I suppose, $\mathbb{F}_{p^k}$ with k > 1 should work.2012-02-20