JOURNAL BROWSE
Search
Advanced SearchSearch Tips
TWO GENERALIZATIONS OF LCM-STABLE EXTENSIONS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
TWO GENERALIZATIONS OF LCM-STABLE EXTENSIONS
Chang, Gyu Whan; Kim, Hwankoo; Lim, Jung Wook;
  PDF(new window)
 Abstract
Let be an extension of integral domains, X be an indeterminate over T, and R[X] and T[X] be polynomial rings. Then is said to be LCM-stable if for all . Let be the so-called -operation on an integral domain A. In this paper, we introduce the notions of - and -LCM-stable extensions: (i) is -LCM-stable if for all and (ii) is -LCM-stable if for all . We prove that LCM-stable extensions are both -LCM-stable and -LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., ), then is -LCM-stable (resp., -LCM-stable) if and only if is -LCM-stable (resp., -LCM-stable).
 Keywords
star-operation;LCM-stable;w-LCM-stable;w(e)-LCM-stable;PvMD;Krull domain;
 Language
English
 Cited by
1.
ON LCM-STABLE MODULES, Journal of Algebra and Its Applications, 2014, 13, 04, 1350133  crossref(new windwow)
 References
1.
T. Akiba, LCM-stableness, Q-stableness and flatness, Kobe J. Math. 2 (1985), no. 1, 67-70.

2.
D. D. Anderson and S. J. Cook, Two star-operations and their induced lattices, Comm. Algebra 28 (2000), no. 5, 2461-2475. crossref(new window)

3.
D. D. Anderson, E. G. Houston, and M. Zafrullah, t-linked extensions, the t-class group, and Nagata's theorem, J. Pure Appl. Algebra 86 (1993), no. 2, 109-124. crossref(new window)

4.
D. F. Anderson and G. W. Chang, Overrings as intersections of localizations of an integral domain, preprint.

5.
G. W. Chang, *-Noetherian domains and the ring $D[X]N_*$, J. Algebra 297 (2006), no. 1, 216-233. crossref(new window)

6.
J. T. Condo, LCM-stability of power series extensions characterizes Dedekind domains, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2333-2341.

7.
D. E. Dobbs, On the criteria of D. D. Anderson for invertible and flat ideals, Canad. Math. Bull. 29 (1986), no. 1, 25-32. crossref(new window)

8.
D. E. Dobbs, E. G. Houston, T. G. Lucas, and M. Zafrullah, t-linked overrings and Prufer v-multiplication domains, Comm. Algebra 17 (1989), no. 11, 2835-2852. crossref(new window)

9.
R. Gilmer, An embedding theorem for HCF-rings, Proc. Cambridge Philos. Soc. 68 (1970), 583-587. crossref(new window)

10.
R. Gilmer, Finite element factorization in group rings, Ring theory, 47-61, Lecture Notes in Pure and Appl. Math., Vol. 7, Dekker, New York, 1974.

11.
R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure Appl. Math. 90, Queen's University, Kingston, Ontario, 1992.

12.
J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18 (1980), no. 1, 37-44. crossref(new window)

13.
E. G. Houston and M. Zafrullah, On t-invertibility. II, Comm. Algebra 17 (1989), no. 8, 1955-1969. crossref(new window)

14.
B. G. Kang, *-operations on integral domains, Ph.D. Dissertation, Univ. Iowa 1987.

15.
B. G. Kang , Prufer v-multiplication domains and the ring $R[X]N_v$, J. Algebra 123 (1989), no. 1, 151-170. crossref(new window)

16.
D. J. Kwak and Y. S. Park, On t-flat overrings, Chinese J. Math. 23 (1995), no. 1, 17-24.

17.
A. Mimouni, Integral domains in which each ideal is a w-ideal, Comm. Algebra 33 (2005), no. 5, 1345-1355. crossref(new window)

18.
S. Oda and K. Yoshida, Remarks on LCM-stableness and reflexiveness, Math. J. Toyama Univ. 17 (1994), 93-114.

19.
J. Sato and K. Yoshida, The LCM-stability on polynomial extensions, Math. Rep. Toyama Univ. 10 (1987), 75-84.

20.
H. Uda, LCM-stableness in ring extensions, Hiroshima Math. J. 13 (1983), no. 2, 357-377.

21.
H. Uda , $G_2$-stableness and LCM-stableness, HiroshimaMath. J. 18 (1988), no. 1, 47-52.

22.
F. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), no. 4, 1285-1306. crossref(new window)

23.
H. Yin, F. Wang, X. Zhu, and Y. Chen, w-modules over commutative rings, J. Korean Math. Soc. 48 (2011), no. 1, 207-222. crossref(new window)

24.
M. Zafrullah, Putting t-invertibility to use, Non-Noetherian commutative ring theory, 429-457, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000.