TITLE: Example of an endomorphism on an abelian group that is not left multiplication
QUESTION [6 upvotes]: It is well-known that all endomorphisms on the abelian group ($\Bbb{Z}$,+) can be seen as a left multiplication by some element in some ring structure on ($\Bbb{Z}$,+); namely left multiplication by any integer in the standard $(\Bbb{Z},+,\times)$ ring.
So far every endomorphism on abelian groups that I have examined has turned out to have this same interesting property, but I'm not very knowledgeable in advanced maths.
Can someone provide an example of an abelian group $G$ with an endomorphism that cannot be seen as a left multiplication by some element in some ring structure on $G$?

REPLY [4 votes]: There are some abelian groups that admit no (possibly nonunital) ring structure with a left unit, and for such groups, the identity endomorphism cannot be multiplication by any element in any ring structure.  The standard example of such a group is $G=\mathbb{Q}/\mathbb{Z}$.  If you had a ring structure on $G$ with in which (the equivalence class of) $u=a/b$ was a left unit for some $a,b\in\mathbb{Z}$, then $bu=0$, so $bx=b(ux)=(bu)x=0$ for all $x\in G$.  But this is clearly false, because (for instance) $b\cdot 1/2b=1/2\neq 0$.