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On the De Morgan’s Laws for Modules

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Abstract

In this investigation we give a module-theoretic counterpart of the well known De Morgan’s laws for rings and topological spaces. We observe that the module-theoretic De Morgan’s laws are related with semiprime modules and modules in which the annihilator of any fully invariant submodule is a direct summand. Also, we give a general treatment of De Morgan’s laws for ordered structures (idiomatic-quantales). At the end, the manuscript goes back to the ring theoretic realm, in this case we study the non-commutative counterpart of Dedekind domains, and we describe Asano prime rings using the strong De Morgan law.

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Acknowledgements

want to thank the referee for his/her comments that improved substantially this manuscript in particular for pointing out Remark 3.10. Part of this investigation was made during a visit of the first author to the Universidad de Guadalajara. He wishes to thank the members of the Department of Mathematics for their kind hospitality. This visit was supported by the program PROSNI 2019 of the third author. The documents [40, 41] were available on the author’s personal web page. Unfortunately these references are not available anymore.

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Authors and Affiliations

  1. Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 Sur, Col.San Manuel, 72570, Ciudad Universitaria, Puebla, Mexico

    Mauricio Medina-Bárcenas

  2. Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340, Ciudad de México, Iztapalapa, Mexico

    Martha Lizbeth Shaid Sandoval-Miranda

  3. Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Blvd. Marcelino García Barragán, 44430, Guadalajara, Jalisco, Mexico

    Ángel Zaldívar-Corichi

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  1. Mauricio Medina-Bárcenas
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  2. Martha Lizbeth Shaid Sandoval-Miranda
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  3. Ángel Zaldívar-Corichi
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Correspondence to Ángel Zaldívar-Corichi.

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Communicated by Jorge Picado.

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Martha Lizbeth Shaid Sandoval-Miranda thanks project PRODEP PTC-2019 Grant UAM-PTC-700 Num. 12613411 awarded by SEP.

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Medina-Bárcenas, M., Sandoval-Miranda, M.L.S. & Zaldívar-Corichi, Á. On the De Morgan’s Laws for Modules. Appl Categor Struct 30, 265–286 (2022). https://doi.org/10.1007/s10485-021-09656-8

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  • DOI: https://doi.org/10.1007/s10485-021-09656-8

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