Every Ring is Isomorphic to a Subring of an Endomorphism Ring of an Abelian Group

You are correct, (madame or) sir.

This is essentially the ring-theoretic analogue of Cayley's Theorem for groups.

Also, this issue (as a question) came up a while back on Math Overflow.

Added: I had missed that the explicit definition of the map was not contained in the OP's question. A natural ring embedding from $R$ to $\operatorname{End}(R,+)$ is

$r \mapsto \bullet r: (x \in R \mapsto xr)$.

[Or possibly $r \mapsto r \bullet: (x \in R \mapsto rx)$, depending upon your conventions on composition.]