I am interested in extensions $A\leq B$ of commutative rings with the property that for all ideals $I\leq A$ we have $IB\cap A=I$. Is there a standard name for this property, or a standard reference for results about it?



Terminology/literature for $\forall I\leq A,\; IB\cap A=I$
Such extension is called "cyclically pure". An extension is called pure if the induced map $A\otimes\_A M\to B\otimes\_A M$ is injective for any $A$ module $M$. If the map $A\to B$ splits as map of $A$-modules, then it is pure, and the converse holds if $B$ is finitely presented as an $A$-module.


Also clearly, purity implies cyclic purity (taking $M=A/I$), and a classic paper by Hochster addressed the converse:
[Cyclic purity versus purity in excellent Noetherian rings](https://www.ams.org/journals/tran/1977-231-02/S0002-9947-1977-0463152-5/S0002-9947-1977-0463152-5.pdf).


