Abstract
We study the existence of maximal ideals in preadditive categories defining an order \(\preceq \) between objects, in such a way that if there do not exist maximal objects with respect to \(\preceq \), then there is no maximal ideal in the category. In our study, it is sometimes sufficient to restrict our attention to suitable subcategories. We give an example of a category \(\mathbf {C}_F\) of modules over a right noetherian ring R in which there is a unique maximal ideal. The category \(\mathbf {C}_F\) is related to an indecomposable injective module F, and the objects of \(\mathbf {C}_F\) are the R-modules of finite F-rank.
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Communicated by Walter P. Tholen.
The first author is partially supported by projects MTM2014-54439 and MTM2016-77445-P from MEC and by research group FQM211 from Junta de Andalucía.
The second author is partially supported by Dipartimento di Matematica, Università di Padova (Progetto SID 2016 BIRD163492/16 “Categorical homological methods in the study of algebraic structures” and progetto DOR1690814 “Anelli e categorie di moduli”).
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Cortés-Izurdiaga, M., Facchini, A. Maximal Ideals in Module Categories and Applications. Appl Categor Struct 26, 617–629 (2018). https://doi.org/10.1007/s10485-017-9505-z
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DOI: https://doi.org/10.1007/s10485-017-9505-z