# Separability of a subring and of a pre-image

Let $A \subseteq B \subseteq C$ be commutative rings.

A known result, which can be found in De Meyer and Ingraham's book, says that separability of $A \subseteq C$ implies separability of $B \subseteq C$.

When separability of $A \subseteq C$ implies separability of $A \subseteq B$? By 'when' I mean what additional condition guarantees that implication.

Another, somewhat similar, question: Let $A \subseteq B$, $A \subseteq C$, $B \twoheadrightarrow C$ be commutative rings.

When separability of $A \subseteq C$ implies separability of $A \subseteq B$?

Thank you very much for any help!

Edit: (1) By separable I mean the definition as in the above mentioned book on page 40 (this page is available in the preview), namely: $A \subseteq B$ is separable if $B$ is a projective $B \otimes_A B$-module. The definition of separability also appears here. (2) There is an answer to a special case of my first question, due to Adjamagbo's transfer theorem. In that special case, 'when'= $B \subseteq C$ is flat, or 'when'= $B$ is normal.

I do not know what definition of separability you are using. $A\subset C$ is separable means every element of $C$ is separable over $A$ in the usual definition. Thus, for any subring $A\subset B\subset C$, clearly $B$ is separable over $A$, since every element of $B$ is an element of $C$ and thus separable.
• If we deal with three fields, then the second question also has a positive answer: By assumption $B \twoheadrightarrow C$ (surjective) and trivially it is injective ($B$ is a field), so $B$ and $C$ are isomorphic, hence separability of $A \subseteq C$ clearly implies separability of $A \subseteq B$. Jul 19, 2017 at 12:49