Non-local ring embedded in a direct product of fields
In this paper we study the immersion of a non-local commutative ring with unity R into a direct productof fields. In the product of quotient fields defined by the maximal ideals of R. The ring homomorphismϕ from R into direct product of quotient fields is defined by the universal property of the dir...
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| Format: | Online |
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Universidad Pedagógica y Tecnológica de Colombia
2024
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| Online Access: | https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/15963 |
| _version_ | 1801706359935205376 |
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| author | Granados Pinzón, Claudia |
| author_facet | Granados Pinzón, Claudia |
| author_sort | Granados Pinzón, Claudia |
| collection | OJS |
| description | In this paper we study the immersion of a non-local commutative ring with unity R into a direct productof fields. In the product of quotient fields defined by the maximal ideals of R. The ring homomorphismϕ from R into direct product of quotient fields is defined by the universal property of the direct product.Let Kerϕ be the kernel of ϕ, then Kerϕ = J (R), with J (R) is the Jacobson radical of the ring R. IfJ (R) = {0}, the ring homomorphism is injective in the infinite case and in the finite case, we will proof ϕis an isomorphism. In addition, we consider R a total ring of fractions with finite number of maximal idealsand will show that the ring homomorphism from R into a direct product of localizations is injective. Evenmore, if R have the form Zn, with n ̸= 0, or R is a finite dimensional K−algebra with field K, we have thatthis ring homomorphism is an isomorphism. |
| format | Online |
| id | oai:oai.revistas.uptc.edu.co:article-15963 |
| institution | Revista Ciencia en Desarrollo |
| publishDate | 2024 |
| publisher | Universidad Pedagógica y Tecnológica de Colombia |
| record_format | ojs |
| spelling | oai:oai.revistas.uptc.edu.co:article-159632024-05-11T07:16:57Z Non-local ring embedded in a direct product of fields Anillo no local inmerso en producto de cuerpos Granados Pinzón, Claudia Anillo total de fracciones, cuerpo cociente, K−álgebra finita, localización, producto directo de anillos, radical de Jacobson. Total ring of fractions, field of fractions, finite dimensional K−algebra, localization, direct product of rings, Jacobson radical. In this paper we study the immersion of a non-local commutative ring with unity R into a direct productof fields. In the product of quotient fields defined by the maximal ideals of R. The ring homomorphismϕ from R into direct product of quotient fields is defined by the universal property of the direct product.Let Kerϕ be the kernel of ϕ, then Kerϕ = J (R), with J (R) is the Jacobson radical of the ring R. IfJ (R) = {0}, the ring homomorphism is injective in the infinite case and in the finite case, we will proof ϕis an isomorphism. In addition, we consider R a total ring of fractions with finite number of maximal idealsand will show that the ring homomorphism from R into a direct product of localizations is injective. Evenmore, if R have the form Zn, with n ̸= 0, or R is a finite dimensional K−algebra with field K, we have thatthis ring homomorphism is an isomorphism. En este artículo estudiamos la inmersión de R, un anillo conmutativo con unidad no local, en un productodirecto de cuerpos. En el producto de los cuerpos cocientes de R dados por sus ideales maximales. Elhomomorfismo ϕ de R en el producto directo de cuerpos cocientes está definido por la propiedad universaldel producto y su núcleo es Kerϕ = J (R), donde J (R) es el radical de Jacobson de R. Si J (R) = {0},el homomorfismo es inyectivo en el caso infinito, y en el caso finito probaremos que ϕ es un isomorfismo.Además, consideramos el caso donde R es un anillo total de fracciones con un número finito de idealesmaximales y mostraremos que el homomorfismo de R en el producto de sus localizados es inyectivo. Másaún, si R es de la forma Zn, con n ̸= 0, o R es una K−álgebra finita, con K un cuerpo, tenemos que estehomomorfismo es un isomorfismo. Universidad Pedagógica y Tecnológica de Colombia 2024-04-09 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Mathematics research article Artículo de investigación en matem´aticas https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/15963 10.19053/01217488.v15.n1.2024.15963 Ciencia En Desarrollo; Vol. 15 No. 1 (2024): Vol 15, Núm.1 (2024): Enero-Junio Ciencia en Desarrollo; Vol. 15 Núm. 1 (2024): Vol 15, Núm.1 (2024): Enero-Junio 2462-7658 0121-7488 |
| spellingShingle | Anillo total de fracciones, cuerpo cociente, K−álgebra finita, localización, producto directo de anillos, radical de Jacobson. Total ring of fractions, field of fractions, finite dimensional K−algebra, localization, direct product of rings, Jacobson radical. Granados Pinzón, Claudia Non-local ring embedded in a direct product of fields |
| title | Non-local ring embedded in a direct product of fields |
| title_alt | Anillo no local inmerso en producto de cuerpos |
| title_full | Non-local ring embedded in a direct product of fields |
| title_fullStr | Non-local ring embedded in a direct product of fields |
| title_full_unstemmed | Non-local ring embedded in a direct product of fields |
| title_short | Non-local ring embedded in a direct product of fields |
| title_sort | non local ring embedded in a direct product of fields |
| topic | Anillo total de fracciones, cuerpo cociente, K−álgebra finita, localización, producto directo de anillos, radical de Jacobson. Total ring of fractions, field of fractions, finite dimensional K−algebra, localization, direct product of rings, Jacobson radical. |
| topic_facet | Anillo total de fracciones, cuerpo cociente, K−álgebra finita, localización, producto directo de anillos, radical de Jacobson. Total ring of fractions, field of fractions, finite dimensional K−algebra, localization, direct product of rings, Jacobson radical. |
| url | https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/15963 |
| work_keys_str_mv | AT granadospinzonclaudia nonlocalringembeddedinadirectproductoffields AT granadospinzonclaudia anillonolocalinmersoenproductodecuerpos |
