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 gen...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203203471 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |