Extensions of rational modules

For a coalgebra C, the rational functor Rat (−):ℳC∗→ℳC∗ is a left exact preradical whose associated linear topology is the family ℱC, consisting of all closed and cofinite right ideals of C∗. It was proved by Radford (1973) that if C is right ℱ-Noetherian (which means that every I∈ℱC is finitely generated), then Rat (−) is a radical. We show that the converse follows if C1, the second term of the coradical filtration, is right ℱ-Noetherian. This is a consequence of our main result on ℱ-Noetherian coalgebras which states that the following assertions are equivalent: (i) C is right ℱ-Noetherian; (ii) Cn is right ℱ-Noetherian for all n∈ℕ; and (iii) ℱC is closed under products and C1 is right ℱ-Noetherian. New examples of right ℱ-Noetherian coalgebras are provided.