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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| Language: | English |
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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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| author | J. Cuadra |
| author_facet | J. Cuadra |
| author_sort | J. Cuadra |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-312b25282d024472b7bdb97931d71a88 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2003-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-312b25282d024472b7bdb97931d71a882025-02-03T01:26:47ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252003-01-012003694363437110.1155/S0161171203203471Extensions of rational modulesJ. Cuadra0Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Almería 04120 , SpainFor 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.http://dx.doi.org/10.1155/S0161171203203471 |
| spellingShingle | J. Cuadra Extensions of rational modules International Journal of Mathematics and Mathematical Sciences |
| title | Extensions of rational modules |
| title_full | Extensions of rational modules |
| title_fullStr | Extensions of rational modules |
| title_full_unstemmed | Extensions of rational modules |
| title_short | Extensions of rational modules |
| title_sort | extensions of rational modules |
| url | http://dx.doi.org/10.1155/S0161171203203471 |
| work_keys_str_mv | AT jcuadra extensionsofrationalmodules |