Locally GCD domains and the ring $D+XD_S[X]$

Document Type : Research Paper


1 Department of Mathematics Education‎, ‎Incheon National University‎, ‎Incheon 406-772‎, ‎Republic of Korea.

2 Facultatea de Matematica si Informatica‎, ‎University of‎ ‎Bucharest‎, ‎14 Academiei Str.‎, ‎Bucharest‎, ‎RO 010014‎, ‎Romania

3 Department of Mathematics‎, ‎Idaho State University‎, ‎Poca-tello‎, ‎ID 83209‎, ‎USA


An integral domain $D$ is called a emph{locally GCD domain} if $D_{M}$ is a GCD domain for every maximal ideal $M$ of $D$. We study some ring-theoretic properties of locally GCD domains. E.g., we show that $D$ is a locally GCD domain if and only if $aD\cap bD$ is locally principal for all $0\neq a,b\in D$, and flat overrings of a locally GCD domain are locally GCD. We also show that the t-class group of a locally GCD domain is just its Picard group. We study when a locally GCD domain is Pr"{u}fer or a generalized GCD domain. We also characterize locally factorial domains as domains $D$ whose minimal prime ideals of a nonzero principal ideal are locally principal and discuss conditions that make them Krull domains. We use the $D+XD_{S}[X]$ construction to give some interesting examples of locally GCD domains that are not GCD domains.


Main Subjects

M. M. Ali, Generalized GCD rings IV, Beitrage Algebra Geom. 55 (2014), no. 2, 371-- 386.
D. D. Anderson, domains, overrings, and divisorial ideals, Glasg. Math. J. 19 (1978), no. 2, 199--203.
D. D. Anderson and D. F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Pauli 28 (1980), no. 2, 215--221.
D. D. Anderson, Globalization of some local properties in Krull domains, Proc. Amer. Math. Soc. 85 (1982), no. 2, 141--145.
D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), no. 1, 78--93.
D. D. Anderson, D. F. Anderson and M. Zafrullah, A generalization of unique factorization, Boll. Un. Mat. Ital. A (7) 9 (1995), no. 2, 401--413.
D. D. Anderson, D. F. Anderson and M. Zafrullah, The ring D+XDS[X] and t-splitting sets, Arab. J. Sci. Eng. Sect. C Theme Issues 26 (2001), no. 1, 3--16.
D. D. Anderson and G. W. Chang and M. Zafrullah, Integral domains of finite t-character, J. Algebra 396 (2013) 169--183.
D. D. Anderson and B. G. Kang, Pseudo-Dedekind domains and divisorial ideals in R[X]T , J. Algebra 122 (1989), no. 2, 323--336.
D. D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), no. 4, 907--913.
D. D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A (7) 8 (1994), no. 3, 397--402.
D. F. Anderson, The class group and local class group of an integral domain, Non-Noetherian commutative ring theory, 33--55, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000.
J. T. Arnold and R. Matsuda, An almost Krull domain with divisorial height one primes, Canad. Math. Bull. 29 (1986), no. 1, 50--53.
A. Bouvier, Le groupe des classes d'un anneau integre, 107eme Congres National des Societes Savantes, Brest, 1982, fasc. IV, 85--92.
J. Brewer and W. Heinzer, Associated primes of principal ideals, Duke Math. J. 41 (1974) 1--7.
P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968) 251--264.
D. Costa, J. Mott and M. Zafrullah, The construction D + XDS[X], J. Algebra 53 (1978), no. 2, 423--439.
P. Eakin and J. Silver, Rings which are almost polynomial rings, Trans. Amer. Math. Soc. 174 (1972) 425--449.
M. Fontana, E. Houston and T. Lucas, Factoring ideals in integral domains, Lecture Notes of the Unione Matematica Italiana, 14. Springer, Heidelberg, Bologna, 2013.
M. Fontana and M. Zafrullah, On v-domains: A survey, Commutative algebra--Noetherian and non-Noetherian Perspectives, 145-179, Springer, New York, 2011.
R. Gilmer, Integral domains which are almost Dedekind, Proc. Amer. Math. Soc. 15 (1964) 813--818.
R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972.
R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974) 65--86.
F. Halter-Koch, Ideal Systems -- An Introduction to Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1998.
J. R. Hedstrom and E. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18 (1980), no. 1, 37--44.
W. Heinzer and J. Ohm, An essential ring which is not a v-multiplication ring Canad. J. Math. 25 (1973) 856--861.
E. Houston and M. Zafrullah, Integral domains in which each t-ideal is divisorial, Michigan Math. J. 35 (1988), no. 2, 291--300.
B. G. Kang, On the converse of a well-known fact about Krull domains, J. Algebra 124 (1989), no. 2, 284--299.
I. Kaplansky, Commutative Rings, Polygonal Publishing House, New Jersey, 1994.
M. D. Larsen and P. J. McCarthy, Multiplicative Theory of Ideals, 43, Academic Press, New York-London, 1971.
S. Malik, J. Mott and M. Zafrullah, On t-invertibility, Comm. Algebra 16 (1988), no. 1, 149--170.
J. Mott and M. Zafrullah, On Prufer v-multiplication domains, Manuscripta Math. 35 (1981), no. 1-2, 1--26.
N. Nakano, Idealtheorie in einem speziellen unendlichen algebraischen Zahlkorper, J. Sci. Hiroshima Univ. Ser. A. 16 (1953) 425--439.
J. Querre, Intersections d'anneaux integres, J. Algebra 43 (1976), no. 1, 55--60.
M. Zafrullah, On a result of Gilmer, J. London Math. Soc. (2) 16 (1977), no. 1, 19--20.
M. Zafrullah, On finite conductor domains, Manuscripta Math. 24 (1978), no. 2, 191--203.
M. Zafrullah, On generalized Dedekind domains, Mathematika 33 (1986), no. 2, 285--296.
M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), no. 9, 1895--1920.
M. Zafrullah, The D+XDS[X] construction from GCD domains, J. Pure Appl. Algebra 50 (1988), no. 1, 93--107.
M. Zafrullah, Ascending chain conditions and star operations, Comm. Algebra 17 (1989), no. 6, 1523--1533.
M. Zafrullah, Flatness and invertibility of an ideal, Comm. Algebra 18 (1990), no. 7, 2151--2158.
M. Zafrullah, Putting t-invertibility to use, Non-Noetherian commutative ring theory, 429--457, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000.
  • Receive Date: 09 July 2014
  • Revise Date: 14 December 2014
  • Accept Date: 16 December 2014
  • First Publish Date: 01 April 2016