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Integral Domains and Their Algebras: Topological Aspects

Authors: Shai Sarussi


Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R∩F = S and the localization of R with respect to S \{0} is A. Denoting by W the set of all S-nice subalgebras of A, and defining a notion of open sets on W, one can view W as a T0-Alexandroff space. Thus, the algebraic structure of W can be viewed from the point of view of topology. It is shown that every nonempty open subset of W has a maximal element in it, which is also a maximal element of W. Moreover, a supremum of an irreducible subset of W always exists. As a notable connection with valuation theory, one considers the case in which S is a valuation domain and A is an algebraic field extension of F; if S is indecomposed in A, then W is an irreducible topological space, and W contains a greatest element.

Keywords: Algebras over integral domains, Alexandroff topology, valuation domains, integral domains.

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[1] P. Alexandroff, Diskrete R¨aume, Mat. Sb. (N.S.) 2 (1937), 501-518.
[2] D. E. Dobbs and M. Fontana, Kronecker function rings and abstract Riemann surfaces, J. Algebra 99 (1986), 263-274.
[3] D. E. Dobbs, R. Fedder, and M. Fontana, Abstract Riemann surfaces of integral domains and spectral spaces, Ann. Mat. Pura Appl. 148 (1987), 101115.
[4] O. Endler, Valuation Theory. Springer-Verlag, New York, 1972.
[5] C. A. Finocchiaro, M. Fontana, and K. A. Loper, The constructible topology on spaces of valuation domains, Trans. Am. Math. Soc. 365 (2013), 6199-6216
[6] R. Huber and M. Knebusch, On valuation spectra, in “Recent advances in real algebraic geometry and quadratic forms: proceedings of the RAGSQUAD year”, Berkeley, 1990-1991, Contemp. Math. 155, Amer. Math. Soc. Providence RI (1994), 167-206.
[7] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60.
[8] F. V. Kuhlmann, Places of algebraic fields in arbitrary characteristic, Advances Math. 188 (2004), 399-424.
[9] S. Sarussi, Quasi-valuations extending a valuation, J. Algebra 372 (2012), 318-364.
[10] S. Sarussi, Quasi-valuations – topology and the weak approximation theorem, Valuation theory in interaction, EMS Series of Congress Reports, EMS Publishing House, 2014, pp. 464-473.
[11] S. Sarussi, Totally ordered sets and the prime spectra of rings, Comm. Algebra, (2017) 45:1, 411-419, DOI: 10.1080/00927872.2016.1175583.
[12] S. Sarussi, Quasi-valuations and algebras over valuation domains, Comm. Algebra, (2019), DOI: 10.1080/00927872.2018.1522322.
[13] S. Sarussi, Extensions of integral domains and quasi-valuations, Comm. Algebra, (2019), DOI: 10.1080/00927872.2019.1677695
[14] S. Sarussi, Alexandroff Topology of Algebras Over an Integral Domain, Mediterr. J. Math., (2020), 1660-5446/20/020001-17
[15] M. H. Stone, The Theory of Representation for Boolean Algebras, Trans. Amer. Math. Soc. 40 (1936) 37-111.
[16] M. H. Stone, Topological representations of distributive lattices and Brouwerian logics, ˇCasopis Peˇst. Mat. Fys. 67 (1937), 1-25.
[17] O. Zariski, The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc. 50, (1944), 683-691.