An example of an endomorphism Could someone suggest a simple $\phi\in $End$_R(A)$ where $A$ is a finitely generated module over ring $R$ where $\phi$ is injective but not surjective? I have a hunch that it exists but I can't construct an explicit example. Thanks.

Consider the morphism of $\mathbb{R}$-modules:
$$
\varphi : \mathbb{R}^\infty \longrightarrow \mathbb{R}^\infty
$$
defined by
$$
\varphi (x_1, x_2, \dots , x_n, \dots ) = (0, x_1, x_2 , \dots , x_n , \dots ) \ .
$$
This example is not possible with finite dimension vector spaces, because then, with endomorphisms, you have
$$
\text{isomorphism} \quad \Longleftrightarrow \quad \text{monomorphism} \quad \Longleftrightarrow \quad \text{epimorphism} \ .
$$
EDIT. Now I see you've added the finitely generated condition. So, this example doesn't apply any more obviously.
