Let $C$ be a symmetric monoidal category. A unital associative algebra $(A,m:A\otimes A\to A)$ is called separable if there exists an $A$-$A$-bimodule homomorphism $d: A\to A\otimes A$ such that $m\circ d =\mathrm{id}_A$.
When we consider the category $\mathrm{Vect}_k$ of $k$-vector spaces for some field $k$ whose symmetric monoidal structure is given by the relative tensor product $\otimes_{k}$, the separable algebras defined above in $\mathrm{Vect}_k$ concides with the separable algebras that appear in separable field extensions over $k$. And for such seprable $k$-algebras, we have a lot of fruitful results.
Now, note that the category $\mathrm{Abel}$ of abelian groups also has a symmetric monoidal structure given by $\otimes_{\mathbb{Z}}$. So it is natural to ask what are separable algebras in $\mathrm{Abel}$. Recall that a unital associative algebra in $\mathrm{Abel}$ is a unital ring. Hence the question is to classify all separable (unital) rings.
Can someone provide some result about this quesion?
What I know is that the integer ring itself is separable, which is obvious, and the field of complex numbers is also separable. And I saw an answer of Classifiation of separable algebras over a commutative ring, maybe we can use the results mentioned in that answer to solve this problem.
Moreover, there is a notion called the "separable ring extension", does this notion has some relation to the separable rings as the separable algebras over a field is equivent to separable extensions?