Abstract
The notion of maximal non-valuative domain is introduced and characterized. An integral domain R is called a maximal non-valuative domain if R is not a valuative domain but every proper overring of R is a valuative domain. Maximal non-valuative domains have at most four maximal ideals. Various properties of maximal non-valuative domains are discussed. We characterize integrally closed maximal non-valuative domains in terms of B\(\acute{\text {e}}\)zout domains. A local non-integrally closed maximal non-valuative domain is also characterized. Conditions are given under which pseudo-valuation domains and maximal non-pseudo-valuation domains are maximal non-valuative domains.
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References
Ayache A and Jaballah A, Residually algebraic pairs of rings, Math. Z. 225 (1997) 49–65
Ben Nasr M and Jarboui N, Maximal non-Jaffard subrings of a field, Publ. Math. 4 (2000) 157–175
Ben Nasr M and Jarboui N, On maximal non-valuation subrings, Houston J. Math. 37(1) (2011) 47–59
Cahen P J, Dobbs D E and Lucas T G, Valuative domains, J. Algebra Appl. 9 (2010) 43–72
Dobbs D E and Fontana M, Universally incomparable ring homomorphisms, Bull. Austral. Math. Soc. 29(3) (1984) 289–302
Ferrand D and Olivier J -P, Homomorphismes minimaux d’anneaux, J. Algebra 16 (1970) 461–471
Gilmer R and Heinzer W, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7 (1967) 133–149
Hedstrom J R and Houston E G, Pseudo-valuation domains, Pacific J. Math. 75(1) (1978) 137–147
Jarboui N and Trabelsi S, Some results about proper overrings of pseudo-valuation domains, J. Algebra Appl. 15(5) (2016) 1650099
Lewis W J, The spectrum of a ring as a partially ordered set, J. Algebra 25 (1973) 419–434
Modica M L, Maximal Subrings, Ph.D. Dissertation, University of Chicago (1975)
Papick I J, Topologically defined classes of going-down domains, Trans. Am. Math. Soc. 219 (1976) 1–37
Acknowledgements
The second author was supported by the Research Initiation Grant Scheme from Birla Institute of Technology and Science Pilani, Pilani, India.
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Communicated by Manoj Kumar Keshari.
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Gaur, A., Kumar, R. Maximal non-valuative domains. Proc Math Sci 134, 13 (2024). https://doi.org/10.1007/s12044-024-00781-7
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DOI: https://doi.org/10.1007/s12044-024-00781-7