Extensions of rational modules. (Q1774360)

Summary: For a coalgebra \(C\), the rational functor \(\text{Rat}(-)\colon{\mathcal M}_{C^*}\to{\mathcal M}_{C^*}\) is a left exact preradical whose associated linear topology is the family \({\mathcal F}_C\), consisting of all closed and cofinite right ideals of \(C^*\). It was proved by \textit{D. E. Radford} [J. Algebra 26, 512-535 (1973; Zbl 0272.16012)] that if \(C\) is right \(\mathcal F\)-Noetherian (which means that every \(I\in{\mathcal F}_C\) is finitely generated), then \(\text{Rat}(-)\) is a radical. We show that the converse follows if \(C_1\), the second term of the coradical filtration, is right \(\mathcal F\)-Noetherian. This is a consequence of our main result on \(\mathcal F\)-Noetherian coalgebras which states that the following assertions are equivalent: (i) \(C\) is right \(\mathcal F\)-Noetherian; (ii) \(C_n\) is right \(\mathcal F\)-Noetherian for all \(n\in\mathbb{N}\); and (iii) \({\mathcal F}_C\) is closed under products and \(C_1\) is right \(\mathcal F\)-Noetherian. New examples of right \(\mathcal F\)-Noetherian coalgebras are provided.