A commutative ring $R$ is said to be an $f-ring$ if every pure ideal is generated by idempotents. (Recall that the ideal $I$ is said to be pure if for each $a\in I$ there is a $b\in I$ such that $ab = a$.) I am looking for some good references for commutative $f-ring$, are there any?
Here is a reference that you might be interested in. Further references are inside the paper.
Note that in the above paper, the definition of pure ideals are not quite the same as the one you give. These two definitions are equivalent, which is proved here (and this is also a good reference for pure ideals).