For a coalgebraC, the rational functorRat (−):ℳC∗→ℳC∗is a left exact preradical whose associated linear topology is the familyℱC, consisting of all closed and cofinite right ideals ofC∗. It was proved by Radford (1973) that ifCis rightℱ-Noetherian (which means that everyI∈ℱCis finitely generated), thenRat (−)is a radical. We show that the converse follows ifC1, the second term of the coradical filtration, is rightℱ-Noetherian. This is a consequence of our main result onℱ-Noetherian coalgebras which states that the following assertions are equivalent: (i)Cis rightℱ-Noetherian; (ii)Cnis rightℱ-Noetherian for alln∈ℕ; and (iii)ℱCis closed under products andC1is rightℱ-Noetherian. New examples of rightℱ-Noetherian coalgebras are provided.