Let $A \subseteq B$ be commutative noetherian domains.
Of course, if $M$ is an $A$-module, then $M \otimes\_A B$ is a $B$-module.
I am curious to know if there exist additional conditions on $A$ and $B$, such that every $B$-module $N$ is necessarily of the form $M \otimes\_A B$ for some $A$-module $M$.


I do not mind to assume one or more of the following additional conditions:
$A$ is a UFD (but I do not want to assume that $B$ is a UFD).
$A$ is regular.
$B$ is a complete intersection ring (but I do not want to assume that $B$ is regular).
$B$ is a faithfully flat $A$-module.
$B$ is a free $A$-module.


I once ran into a paper (unfortunately I cannot find it now) which calls such $N$ extendable
(maybe that paper answers my curiosity?).



When every module is a scalar extension?
An example: if $B$ is the henselization of a local ring $A$, then for any finite type $B$-module $N$ there exists a finite type $A$-module $M$ such that $N$ is a direct summand of $M\otimes\_AB$. You can find this result and some similar others in <http://arxiv.org/pdf/0707.4197v3.pdf>


I think there are more recent papers on this question maybe by some of the same authors, but I cannot remmenber more precisely.


