Extensions of Rational Modules
نویسنده
چکیده
For a coalgebraC , the rational functor Rat(−) : C∗ → C∗ is a left exact preradical whose associated linear topology is the family C , consisting of all closed and cofinite right ideals of C∗. It was proved by Radford (1973) that if C is right Noetherian (which means that every I ∈ C 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 -Noetherian. This is a consequence of our main result on Noetherian coalgebras which states that the following assertions are equivalent: (i) C is right -Noetherian; (ii) Cn is right -Noetherian for all n∈N; and (iii) C is closed under products and C1 is right -Noetherian. New examples of right Noetherian coalgebras are provided.
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