First, a convention: given an abelian group $X$, write $\mathrm{End}_{\mathbf{Ab}}(X)$ for the set of all group homomorphisms $$X \rightarrow X.$$
Now let $R$ denote a ring.
Question. Given an $R$-module $X$, what do we call the corresponding ring homomorphism $f:R \rightarrow \mathrm{End}_{\mathbf{Ab}}(X)$ given as follows? $$f(r) = \mathop{\lambda}_{x:X} rx$$
The "representor" of $X$, perhaps?