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On strongly primary monoids and domains

A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence, one-dimensional local Mori domains are strongly primary. We prove among othe...

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Detalles Bibliográficos
Autores principales: Geroldinger, Alfred, Roitman, Moshe
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Taylor & Francis 2020
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7446044/
https://www.ncbi.nlm.nih.gov/pubmed/32939190
http://dx.doi.org/10.1080/00927872.2020.1755678
Descripción
Sumario:A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence, one-dimensional local Mori domains are strongly primary. We prove among other results that if R is a domain such that the conductor [Image: see text] vanishes, then [Image: see text] is finite; that is, there exists a positive integer k such that each nonzero nonunit of R is a product of at most k irreducible elements. Using this result, we obtain that every strongly primary domain is locally tame, and that a domain R is globally tame if and only if [Image: see text] In particular, we answer Problem 38 of the open problem list by Cahen et al. in the affirmative. Many of our results are formulated for monoids.