Let $R$ be a commutative ring. Let $M$ be an $R-$module generated by $m_1, \ldots, m_n$. Is $\textrm{End}_R(M)$ a finitely generated $R-$module?
My own attempt at proving this used the endomorphisms
$$ f_{i,j}: f_{i,j}(m_i) = m_j, f_{i,j}(m_k) = 0 (k \neq i),$$
of which I am not sure they are well-defined.
EDIT: Those generating endomorphisms are in general not well-defined; can we impose a condition on $m_1, \ldots, m_n$ so that they are? In a vector space, demanding $m_1, \ldots, m_n$ to be linearly independent would be sufficient.