Basis of a free module under module homomorphisms

Question

Hello people my question is the following: Let R be a principal ideal domain, F be a free R-module with basis {e1,…,eN} and f:F→F a module homomorphism.The R-submodule imf of F is always free since we are working over PID. Is there any case that the set {f(e1),…,f(eN)} does not form a base of imf? It is obvious that the images of the basis element through f generate Imf, so all result to whether these elements could be linearly dependent over R. thanks in advance!

Answer 1

Let $R=k$ be a field, $F=k^3$, $e_i$ the standard basis, $f=\left(\begin{smallmatrix}1 & 0 & 1\\0 & 1 & 1\\0 & 0 & 0\end{smallmatrix}\right)$. Then the $f(e_i)$ are no basis of the image of $f$ since they are linearly dependent.