In this paper, the authors prove several results pertaining to ideals which are liftings of idempotents. In particular,

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every ideal in the nilradical is a lifting of idempotents,

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if \(I \subseteq J\) and Max\((R/J) \rightarrow \text{Max}(R/I)\) is surjective then \(I\) is a lifting of idempotents if \(J\) is,

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if \(I\) is a lifting of idempotents and \(J\) is regular, then \(I+J\) is a lifting of idempotents,

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every ideal is a lifting of idempotents if and only if \(R\) is a clean ring,

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if \(R\) is complete with respect to the \(I\)-adic topology, then \(I\) is a lifting of idempotents,

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if \(I\) and \(J\) are not coprime, \(I\) is a lifting of idempotents and \(R/J\) has no nontrivial idempotents, then \(I \cap J\) is a lifting of idempotents.

In addition, they also prove \(|R| \leq |\text{Spec}(R)|\) for Boolean rings and discuss instances when equality holds. There are also several nice results regarding primitive and orthogonal idempotents. A short appendix includes a proof of the Chinese Remainder Theorem using their results.
The paper includes several concrete examples some which clarify why converses of some of their results do not hold.