  We revisit the concept of a minimal basis through the lens of the theory of
modules over a commutative ring $R$. We first review the conditions for the
existence of a basis for submodules of $R^n$ where $R$ is a B\'{e}zout domain.
Then, we define the concept of invertible basis of a submodule of $R^n$ and,
when $R$ is an elementary divisor domain, we link it to the Main Theorem of [G.
D. Forney Jr., SIAM J. Control 13, 493--520, 1975]. Over an elementary divisor
domain, the submodules admitting an invertible basis are precisely the free
pure submodules of $R^n$. As an application, we let $\Omega \subseteq
\mathbb{C}$ be either a connected compact set or a connected open set, and we
specialize to $R=\mathcal{A}$, the ring of functions that are analytic on
$\Omega$. We show that, for any matrix $A(z) \in \mathcal{A}^{m \times n}$,
$\mathrm{ker} \ A(z) \cap \mathcal{A}^n$ is a free $\mathcal{A}$-module and
admits an invertible basis, or equivalently a basis that is full rank upon
evaluation at any $\lambda \in \Omega$. Finally, given $\lambda \in \Omega$, we
use invertible bases to define and study maximal sets of root vectors at
$\lambda$ for $A(z)$. This in particular allows us to define eigenvectors also
for analytic matrices that do not have full column rank.
