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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Bibliographic Details
Main Author:	Granados Pinzón, Claudia
Format:	Online
Published:	Universidad Pedagógica y Tecnológica de Colombia 2024
Subjects:	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.
Online Access:	https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/15963
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Summary:	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.