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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?

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    $\begingroup$ Just a remark: if $B$ is flat over $A$, this is equivalent to $B$ being faithfully flat over $A$. But of course it can happen in more general situations. $\endgroup$
    – abx
    Commented Feb 16, 2022 at 9:07

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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.

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