Abstract
For a ring extension \({R \subset S, \,(R, S)}\) is called a principal ideal domain pair (for short PID pair) if every domain \({T, \,R \subseteq T \subseteq S}\), is a principal ideal domain. When R is a field it is shown that (R, S) is a PID pair iff S is algebraic over R. When R is not a field it is proved that the only PID pairs are those such that R is a PID and S is an overring of R. The second purpose of this paper is to study maximal non-PID subrings. We characterize these type of rings. Further applications and results are also presented.
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References
Anderson D.F., Dobbs D.E.: Pairs of rings with the same prime ideals. Can. J. Math. 32, 362–384 (1980)
Ayache A., Ben Nasr M., Echi O., Jarboui N.: Universally catenarian and going-down pairs of rings. Math. Z. 238, 695–731 (2001)
Ayache A., Dobbs D.E., Echi O.: On maximal non-ACCP subrings. J. Algebra Appl. 6(5), 873–894 (2007)
Ayache A., Jaballah A.: Residually algebraic pairs of rings. Math. Z. 225, 49–65 (1997)
Ayache A., Jarboui N.: Maximal non-Notherian subring of a domain. J. Algebra 248, 806–823 (2002)
Ben Nasr M.: Some remarks on residually algebraic pairs of rings. Arch. Math. 78, 362–368 (2002)
Ben Nasr M., Jarboui N.: Maximal non-Jaffard subrings of a field. Publ. Math. 44, 157–175 (2000)
Cahen P.-J.: Couple d’anneaux partageant un idéal. Arch. Math. 51, 505–514 (1988)
Ferrand D., Olivier J.: Homomorphismes minimaux d’anneaux. J. Algebra 16, 461–471 (1970)
Fossum R.M.: The Divisor Class Group of a Krull Domain. Springer-Verlag, New York (1973)
Gilmer R.: Multiplicative Ideal Theory. Dekker, New York (1972)
Gilmer R., Heinzer W.: Intersections of quotient rings of an integral domain. J. Math. Kyoto Univ. 7(2), 133–150 (1967)
Hedstrom J.R., Houston E.G.: Pseudo-valuation domains. Pac. J. Math. 75, 137–147 (1978)
Kaplansky I.: Commutative Rings (Revised edition). Univeristy of Chicago University Press, Chicago (1974)
Papick I.J.: A note on proper overrings. Comment. Math. Univ. St. Paul. 38, 137–140 (1979)
Wadsworth A.R.: Pairs of domains where all intermediate domains are Noetherian. Trans. Am. Math. Soc. 195, 201–211 (1974)
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Ayache, A., Ben Nasr, M. & Jarboui, N. PID pairs of rings and maximal non-PID subrings. Math. Z. 268, 635–647 (2011). https://doi.org/10.1007/s00209-010-0687-4
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DOI: https://doi.org/10.1007/s00209-010-0687-4