Some adjunctions associated with extensions and restrictions of ideals in the context of commutative rings
Given a commutative ring R and S one of its ideals, the function I -- and gt; (I : S) that transforms ideals of R into ideals of R, is right adjoint of the function I -- and gt; IS. We define the S−maximal ideals of R as those ideals J of R such that (J : S) = J. If the ring S is pseudo-regular, the...
- Autores:
-
Acosta, Lorenzo
Rubio, Marcela
- Tipo de recurso:
- Article of journal
- Fecha de publicación:
- 2013
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/73886
- Acceso en línea:
- https://repositorio.unal.edu.co/handle/unal/73886
http://bdigital.unal.edu.co/38363/
- Palabra clave:
- Ideal
Prime ideal
Semi-prime ideal
Ordered set
Adjoint functions.
Ideal
ideal primo
ideal semi-primo
conjunto ordenado
funciones adjuntas.
- Rights
- openAccess
- License
- Atribución-NoComercial 4.0 Internacional
| Summary: | Given a commutative ring R and S one of its ideals, the function I -- and gt; (I : S) that transforms ideals of R into ideals of R, is right adjoint of the function I -- and gt; IS. We define the S−maximal ideals of R as those ideals J of R such that (J : S) = J. If the ring S is pseudo-regular, then the set of S−maximal ideals of R is a complete lattice, isomorphic to the lattice of the ideals of S. In particular, the annihilator of S in R is the minimum of the S−maximal ideals of R. So the lattice structure of S−maximal ideals of R does not depend on the ring R.On the other hand, the ideals of S can be extended to ideals of R and the ideals of R can be restricted to ideals of S. These two processes are not adjoint to each other, but if we restrict to appropriated collections of ideals we can obtain adjunctions. |
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