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Abstractly constructed prime spectra

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The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum \({\mathrm {Spec}}(R)\) of a unital commutative ring R is always a spectral (= coherent) topological space. In this generalization, which includes several other known ones, the role of ideals of R is played by elements of an abstract complete lattice L equipped with a binary multiplication with \(xy\leqslant x\wedge y\) for all \(x,y\in L\). In fact when no further conditions on L are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral. This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of L; we also make short additional remarks on semiprime elements. We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world. The cases of groups and of non-commutative rings are briefly considered separately.

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Acknowledgements

We are very grateful to Aldo Ursini and Paolo Aglianò for suggesting to look at the papers [4, 72], and [5].

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Authors and Affiliations

  1. Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    Alberto Facchini

  2. Dipartimento di Matematica e Informatica, Università di Catania, Città Universitaria, viale Andrea Doria 6, 95125, Catania, Italy

    Carmelo Antonio Finocchiaro

  3. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7700, South Africa

    George Janelidze

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  1. Alberto Facchini
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Correspondence to George Janelidze.

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A. Facchini and C. Finocchiaro are supported by the research project “Reducing complexity in algebra, logic, combinatorics” (Fondazione Cariverona). A. Facchini is also partially supported by Progetto di Ricerca di Rilevante Interesse Nazionale “CARTHA”, and C. Antonio by GNSAGA and the projects PIACERI “PLGAVA-Proprietà locali e globali di anelli e di varietà algebriche” and “MTTAI - Metodi topologici in teoria degli anelli e loro ideali” of the University of Catania.

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Facchini, A., Finocchiaro, C.A. & Janelidze, G. Abstractly constructed prime spectra. Algebra Univers. 83, 8 (2022). https://doi.org/10.1007/s00012-021-00764-z

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