Abstract
Some new equivalences to the chain conjecture (the integral closure of a local domain is catenary) and to the catenary chain conjecture (the integral closure of a catenary local domain is catenary) are proved, as are some new characterizations of a local H-domain. Also, a fact which lends support to the chain conjecture is noted, and it is proved that the H-conjecture (a local H-domain is catenary) implies the catenary chain conjecture.
Research on this paper was supported in part by the National Science Foundation, Grant 28939.
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Bibliography
I. Kaplansky, Adjacent prime ideals, J. Algebra 20(1972), 94–97.
W. Krull, Beiträge zur Arithmetik kommutativer Integritätsbereiche. III Zum Dimensionsbegriff der Idealtheorie, Math. Zeit. 42(1937), 745–766.
M. Nagata, On the chain problem of prime ideals, Nagoya Math. J. 10(1956), 51–64.
M. Nagata, Local Rings, Interscience Tracts 13, Interscience, New York, 1962.
L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals, (I), Amer. J. Math. 91(1969), 508–528.
L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals, (II), Amer. J. Math. 92(1970), 99–144.
L. J. Ratliff, Jr., Characterizations of catenary rings, Amer. J. Math. 93(1971), 1070–1108.
L. J. Ratliff, Jr., Catenary rings and the altitude formula, Amer. J. Math., (forthcoming).
J. Sally, Failure of the saturated chain condition in an integrally closed domain, Abstract 70T-A72, Notices Amer. Math. Soc. 17(1970), 560.
O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Van Nostrand, New York, 1960
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© 1973 Springer-Verlag
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Ratliff, L.J. (1973). Chain conjectures and H-domains. In: Brewer, J.W., Rutter, E.A. (eds) Conference on Commutative Algebra. Lecture Notes in Mathematics, vol 311. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0068931
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DOI: https://doi.org/10.1007/BFb0068931
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