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?
2026-04-25 21:12:21.1777151541
Endomorphism ring which is not the endomorphism ring of an abelian group.
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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.
For a more elementary example, neither $\mathbb{F}_p^2$ nor $\mathbb{F}_{p^k}$ are endomorphism rings. If they were, then the underlying ring should be an $\mathbb{F}_p$-vector space; but no vector space has those groups as its only endomorphisms.