KAPLANSKY-TYPE THEOREMS IN GRADED INTEGRAL DOMAINS

Title & Authors
KAPLANSKY-TYPE THEOREMS IN GRADED INTEGRAL DOMAINS
CHANG, GYU WHAN; KIM, HWANKOO; OH, DONG YEOL;

Abstract
It is well known that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero principal prime. This is the so-called Kaplansky`s theorem. In this paper, we give this type of characterizations of a graded PvMD (resp., G-GCD domain, GCD domain, $\small{B{\acute{e}}zout}$ domain, valuation domain, Krull domain, $\small{{\pi}}$-domain).
Keywords
Kaplansky-type theorem;upper to zero;prime (primary) element;graded PvMD;graded GCD domain;graded G-GCD domain;graded $\small{B{\acute{e}}zout}$ domain;graded valuation domain;graded Krull domain;graded $\small{{\pi}}$-domain;
Language
English
Cited by
References
1.
D. D. Anderson and D. F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Paul. 28 (1979), 215-221.

2.
D. D. Anderson and D. F. Anderson, Divisibility properties of graded domains, Canad. J. Math. 34 (1982), no. 1, 196-215.

3.
D. D. Anderson and D. F. Anderson, Divisorial ideals and invertible ideals in a graded integral domain, J. Algebra 76 (1982), no. 2, 549-569.

4.
D. D. Anderson, T. Dumitrescu, and M. Zafrullah, Almost splitting sets and AGCD domains, Comm. Algebra 32 (2004), no. 1, 147-158.

5.
D. D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A (7) 8 (1994), no. 3, 397-402.

6.
D. D. Anderson and M. Zafrullah, On t-invertibility, IV, in Factorization in integral domains (Iowa City, IA, 1996), 221-225, Lecture Notes in Pure and Appl. Math., 189, Dekker, New York, 1997.

7.
D. F. Anderson and G. W. Chang, Homogeneous splitting sets of a graded integral domain, J. Algebra 288 (2005), no. 2, 527-554.

8.
D. F. Anderson and G. W. Chang, Almost splitting sets in integral domains II, J. Pure Appl. Algebra 208 (2007), no. 1, 351-359.

9.
D. F. Anderson and G. W. Chang, Graded integral domains and Nagata rings, J. Algebra 387 (2013), 169-184.

10.
D. F. Anderson, G. W. Chang, and J. Park, Generalized weakly factorial domains, Houston J. Math. 29 (2003), no. 1, 1-13.

11.
G. W. Chang, Almost splitting sets in integral domains, J. Pure Appl. Algebra 197 (2005), no. 1-3, 279-292.

12.
G. W. Chang, B. G. Kang, and J. W. Lim, Prufer v-multiplication domains and related domains of the form D + $D-S[{\Gamma}^*]$, J. Algebra 323 (2010), no. 11, 3124-3133.

13.
G. W. Chang and H. Kim, Kaplansky-type theorems II, Kyungpook Math. J. 51 (2011), no. 3, 339-344.

14.
R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.

15.
R. Gilmer, J. Mott, and M. Zafrullah, t-invertibility and comparability, in Commutative ring theory (Fs, 1992), 141-150, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.

16.
E. G. Houston, S. B. Malik, and J. L. Mott, Characterizations of *-multiplication domains, Canad. Math. Bull. 27 (1984), no. 1, 48-52.

17.
J. L. Johnson, The graded ring R[$X_1$, . . . ,$X_n$], Rocky Mountain J. Math. 9 (1979), no. 3, 415-424.

18.
B. G. Kang, On the converse of a well-known fact about Krull domains, J. Algebra 124 (1989), no. 2, 284-299.

19.
I. Kaplansky, Commutative Rings, Revised Ed., Univ. of Chicago, Chicago, 1974.

20.
H. Kim, Kaplansky-type theorems, Kyungpook Math. J. 40 (2000), no. 1, 9-16.

21.
D. G. Northcott, Lessons on Rings, Modules, and Multiplicities, Cambridge Univ. Press, Cambridge, 1968.

22.
M. Zafrullah, On finite conductor domains, Manuscripta Math. 24 (1978), no. 2, 191-204.

23.
M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), no. 1-3, 29-62.