Extensions of rational modules

J. Cuadra

International Journal of Mathematics and Mathematical Sciences, 2003, vol. 2003, 1-9

Abstract:

For a coalgebra C , the rational functor Rat ( − ) : ℳ C ∗ → ℳ C ∗ is a left exact preradical whose associated linear topology is the family ℱ C , 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 ∈ ℱ C is finitely generated), then Rat ( − ) is a radical. We show that the converse follows if C 1 , 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) C n is right ℱ -Noetherian for all n ∈ ℕ ; and (iii) ℱ C is closed under products and C 1 is right ℱ -Noetherian. New examples of right ℱ -Noetherian coalgebras are provided.

Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:505429

DOI: 10.1155/S0161171203203471

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