International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 69, Pages 4363-4371
doi:10.1155/S0161171203203471
Extensions of rational modules
Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Almería 04120 , Spain
Received 30 March 2002
Copyright © 2003 J. Cuadra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.