Let A be a commutative ring with identity and I an ideal of A. A is said to be I-clean if for every element a ∈ A there is an idempotent e = e 2 ∈ A such that a−e is a unit and ae belongs to I. A filter of ideals, say F, of A is Noetherian if for each I ∈ F there is a finitely generated ideal J ∈ F such that J ⊆ I. We characterize I-clean rings for the ideals 0, n(A), J(A), and A, in terms of the frame of multiplicative Noetherian filters of ideals of A, as well as in terms of more classical ring properties
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