Abstract
Motivated by recent results on commutative rings with zero divisors [2, 11], we investigate the difference between the three notions of locally classical, maximally classical, and classical rings. Motivated also by results in [12], we explore these notions when restricted to certain subsets of the prime spectrum of the ring. As an application, we examine the case of locally classical rings of continuous functions, the case of maximally classical and classical rings having already been considered [1, 14].
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B. Banerjee, S. Ghosh, M. Henriksen, Unions of minimal prime ideals in rings of continuous functions on compact spaces. Algebr. Univers. 62(2–3), 239–246 (2009)
J. Boynton, Prüfer conditions and the total quotient ring. Commun. Algebra 39(5), 1624–1630 (2011)
G. Cherlin, M. Dickmann, Real closed rings I. Residue rings of rings of continuous functions. Fund. Math. 126(2), 147–183 (1986)
F. Dashiell, A. Hager, M. Henriksen, Order-Cauchy completions of rings and vector lattices of continuous functions. Canad. J. Math. 32(3), 657–685 (1980)
N. Fine, L. Gillman, Extensions of continuous functions in \(\beta \mathbb{N}\). Bull. Am. Math. Soc. 66, 376–381 (1960)
L. Gillman, M. Henriksen, Concerning rings of continuous functions. Trans. Am. Math. Soc. 77, 340–362 (1954)
L. Gillman, M. Henriksen, Rings of continuous functions in which every finitely generated ideal is principal. Trans. Am. Math. Soc. 82, 366–391 (1956)
L. Gillman, M. Jerison, Rings of Continuous Functions. Graduate Texts in Mathemetics, vol. 43 (Springer, Berlin, 1976)
R. Gilmer, Multiplicative Ideal Theory. Queen’s Papers in Pure and Applied Mathematics, vol. 90 (Queen’s University, Kingston, 1992)
J. Huckaba, Commutative Rings with Zero Divisors. Monographs and Textbooks in Pure and Applied Mathematics, vol. 117 (Marcel Dekker, New York, 1988)
L. Klingler, T. Lucas, M. Sharma, Maximally Prüfer rings. Commun. Algebra 43, 120–129 (2015)
L. Klingler, T. Lucas, M. Sharma, Local types of Prüfer rings. J. Algebra Appl. To appear
M. Knebusch, D. Zhang, Manis Valuations and Prüfer Extensions I. A New Chapter in Commutative Algebra. Lecture Notes in Mathematics, vol. 1791 (Springer, Berlin, 2002)
R. Levy, Almost P-spaces. Canad. J. Math. 29(2), 284–288 (1977)
J. Martinez, S. Woodward, Bézout and Prüfer f-rings. Commun. Algebra 20(10), 2975–2989 (1992)
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Klingler, L., McGovern, W.W. (2019). Local Types of Classical Rings. In: Badawi, A., Coykendall, J. (eds) Advances in Commutative Algebra. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-7028-1_8
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DOI: https://doi.org/10.1007/978-981-13-7028-1_8
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