Let $M_R$ be a module over a ring $R$ with unit and $f\in \text{End}_R(M)$. If $\ker(f)\cong M/f(M)$, then is it right to conclude that $\ker(f)$ is a finitely generated sub-module of $M$?
My thinking is that $\ker(f)$ is a finitely generated module since each $m\in\ker(f)$ corresponds to some element $m+f(M)\in M/f(M)$. Moreover, each $m+f(M)$ can be written as a linear combination of $m_1+f(M), m_2+f(M),\ldots,m_n+f(M)$.
Another reason is $M/f(M)$ is a finite module. So it is finitely generated.