Zariski-like topologies for lattices with applications to modules over associative rings. (English) Zbl 1430.06005

If \(\mathbb X = \langle X; \tau\rangle\) is a \(T_0\) topological space, and \(K\) is the lattice of closed subsets of \(X\), then the dual lattice \(L=K^{\partial}\) is a complete lattice which has a subset \(\tilde{X}=\{\{p\}\;|\;p\in X\}\subseteq L\) which, when equipped with the Zariski topology, is a space naturally homeomorphic to \(\mathbb X\). That is, the closed subsets of \(\tilde{X}\) in the Zariski topology are the sets \(V(a) = \{\{p\}\in \tilde{X}\;|\;a\leq_L \{p\}\} = \{\{p\}\in \tilde{X}\;|\;p\in a\}\), and the map \(p\mapsto \{p\}\) is a homeomorphism from \(\mathbb X\) to \(\tilde{\mathbb X}\).
This paper starts with an arbitrary complete lattice \(L\) and a proper subset \(X\subsetneq L\) and creates a space on \(X\) with the sets \(V(a) = \{p\in X\;|\;a\leq_L p\}\). \(L\) is called \(X\)-top if the collection of \(V(a)\)’s is closed under union, in which case they form the closed sets of a topology on \(X\). The paper studies the topological properties of the resulting space (separation axioms, connectedness, and compactness), and provides sufficient conditions for the space to be spectral.


06B30 Topological lattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
16D10 General module theory in associative algebras
54B99 Basic constructions in general topology
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[1] Abuhlail, J.; Lomp, C., On topological lattices and an application to module theory, J. Algebra Appl., 15, 3, 1650046, (2016) · Zbl 1343.06005
[2] Abuhlail, J., Zariski topologies for coprime and second submodules, Algebra Colloq., 22, 1, 47-72, (2015) · Zbl 1312.16002
[3] Abuhlail, J., A Zariski topology for modules, Commun. Algebra, 39, 11, 4163-4182, (2011) · Zbl 1253.16043
[4] Abuhlail, J., A dual Zariski topology for modules, Topology Appl., 158, 3, 457-467, (2011) · Zbl 1226.16006
[5] Abuhlail, J., A Zariski topology for bicomodules and corings, Appl. Categorical Struct., 16, 1-2, 13-28, (2008) · Zbl 1182.16025
[6] Abuhlail, J., Fully coprime comodules and fully coprime corings, Applied Categorical Structures, 14, 5-6, 379-409, (2006) · Zbl 1121.16030
[7] Abuhlail, J.; Lomp, C., On the notion of strong irreducibility and its dual, J. Algebra Appl., 12, 6, 1350012, (2013) · Zbl 1286.16001
[8] S. Annin, Associated and attached primes over noncommutative rings, Ph.D. dissertation, University of California at Berkeley (2002). · Zbl 1010.16025
[9] Atiyah, M.; Macdonald, I., Introduction to Commutative Algebra, (1969), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. · Zbl 0175.03601
[10] Behboodi, M.; Haddadi, M. R., Classical Zariski topology of modules and spectral spaces, Int. Elec. J. Algebra, 4, 104-130, (2008) · Zbl 1195.16002
[11] Behboodi, M.; Haddadi, M. R. M., Classical Zariski topology of modules and spectral spaces, II, Int. Elec. J. Algebra, 4, 131-148, (2008) · Zbl 1195.16003
[12] Bourbaki, N., General Topology, (1995), Springer-Verlag: Springer-Verlag, Berlin, Heidelberg · Zbl 0145.19302
[13] Dauns, J., Prime modules, J. Reine Angew. Math., 298, 156-181, (1978) · Zbl 0365.16002
[14] Gratzer, G., Lattice Theory: Foundations, (2010), Birkhäuser: Birkhäuser, Basel
[15] Hochster, M., Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142, 43-60, (1969) · Zbl 0184.29401
[16] McCasland, R. L.; Smith, P. F., Zariski spaces of modules over arbitrary rings, Commun. Algebra, 34, 11, 3961-3973, (2006) · Zbl 1168.16027
[17] McCasland, R.; Moore, M.; Smith, P., On the spectrum of a module over a commutative ring, Commun. Algebra, 25, 79-103, (1997) · Zbl 0876.13002
[18] J. S. Milne, Group Theory, Lecture Notes (2017). http://www.jmilne.org/math/CourseNotes/GT.pdf.
[19] Shick, P. L., Topology, Point-Set and Geometric, (2007), Wiley: Wiley, Hoboken, New Jersey · Zbl 1120.54001
[20] Yassemi, S., The dual notion of prime submodules, Arch. Math. (Brno), 37, 273-278, (2001) · Zbl 1090.13005
[21] I. Wijayanti, Coprime modules and comodules, Ph.D. dissertation, Heinrich-Heine Universität, Düsseldorf (2006).
[22] Wisbauer, R., Algebra, Logic and Applications, 3, Foundations of module and ring theory. A handbook for study and research, (1991), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers, Philadelphia, Pennsylvania
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