Abstract

GRANADOS-PINZON, Claudia; CONTRERAS-MENDOZA, Astrid L.  and  OLAYA-LEON, Wilson. Non-local ring embedded in a direct product of fields. Ciencia en Desarrollo [online]. 2024, vol.15, n.1, pp.97-103.  Epub Oct 15, 2024. ISSN 0121-7488.  https://doi.org/10.19053/01217488.v15.n1.2024.15963.

In this paper we study the immersion of a non-local commutative ring with unity R into a direct product of fields. In the product of quotient fields defined by the maximal ideals of R. The ring homomorphism φ 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 = (R), where (R) is the Jacobson radical of the ring R. If (R) = {0}, the ring homomorphism is injective in the infinite case and in the finite case, we will proof φ is an isomorphism. In addition, we consider R a total ring of fractions with finite mimber of maximal ideals and show that the ring homomorphism from R into a direct product of localizations is injective. Even more, if R have the form ℤn, with n ≠ 0, or R is a finite dimensional K-algebra with K a field, we have that this ring homomorphism is an isomorphism.

Keywords : Total ring of fractions; field of fractions; finite dimensional K-algebra; localization; direct product of rings; Jacobson radical.

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