Let $\mathcal{A}$ be a commutative algebra and let $Der(\mathcal{A})$ be its associated algebra of derivations ( $Der(\mathcal{A})$ is assumed to be non-empty).
A covariant derivative is a map $$\nabla: Der(\mathcal{A}) \times Der(\mathcal{A}) \to Der(\mathcal{A})$$
denoted by $$\nabla_XY$$ that satisfies the properties:
a. It is tensorial ($\mathcal{A}$-linear) in the first (index argument)
b. It is $\mathbb{R}$-linear in the second argument
c. It satisfies the product rule in the second argument: $$ \nabla_X(fY)=X(f)Y+f(\nabla_XY)$$ for all $X,Y \in Der(\mathcal{A})$ and for all $f \in \mathcal{A}$
If $Der(\mathcal{A})$ is a free module one can always construct a covariant derivative on $Der(\mathcal{A}).$ If $Der(\mathcal{A})$ is the Lie Algebra of smooth vector fields on a a smooth manifold one can always define $\nabla$ using a partition of unity. Could this be done for a generic $Der(\mathcal{A})?$