DIVISIBILITY PROPERTIES RELATED TO STAR-OPERATIONS ON INTEGRAL DOMAINS
An integral domain R is a GCD-Bezout domain if the Bezout identity holds for any finite set of nonzero elements of R whose gcd exists. Such domains are characterized as the DW-domains having the PSP-property. Using the notion of primitive and superprimitive ideals, we define a (semi)star operation, the q-operation, which is closely related to the w-operation and the p-operation introduced by Anderson. We use q-operation to characterize the GCD-Bezout domains and study various properties of these domains.
Keywords:
GCD, star operation,
___
- J. Arnold and P. Sheldon, Integral Domains that satisfy Gauss’s Lemma, Michi- gan Math. J., 22 (1975), 39–51.
- D.F. Anderson, Integral v-ideals, Glasgow Math. J., 22 (1981), 167–172.
- A. Bouvier, Le Groupe des Classes d’un anneau int´egre, IV Congr`es national des Soci´et´es Savantes, Brest, 107 (1982), 85–92.
- S. El Baghdadi and S. Gabelli, w-divisorial domains, J. Algebra, 285(1) (2005), –355.
- M. Fontana and S. Gabelli, Pr¨ufer domains with class group generated by the classes of the invertible maximal ideals, Comm. Algebra, 25(12) (1997), 3993–
- M. Fontana and S. Gabelli, On the class group and the local class group of a pullback, J. Algebra, 181(3) (1996), 803–835.
- M. Fontana and J. Huckaba, Localizing systems and semistar operations, Non- Noetherian Commutative Ring Theory, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000, 169–197.
- S. Gabelli and F. Tartarone, On the class group of integer-valued polynomial rings over Krull domains, J. Pure Appl. Algebra, 149 (2000), 47–67.
- R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972; rep.
- Queen’s Papers in Pure and Applied Mathematics, Vol. 90, Queen’s University, Kingston, 1992.
- S. Glaz and W.V. Vasconcelos, Flat ideals. II, Manuscripta Math., 22(4) (1977), 325–341.
- B.G. Kang, Pr¨ufer v-multiplication domains and the ring R[X]Nv, J. Algebra, (1) (1989), 151–170.
- I. Kaplansky, Commutative Rings, Rev. ed. University of Chicago Press, Chicago and London, 1974.
- K.A. Loper, Two Pr¨ufer domain counterexamples, J. Algebra, 221(2) (1999), –643.
- A. Mimouni, Integral domains in which each ideal is a w-ideal, Comm. Algebra, (5) (2005), 1345–1355.
- A. Okabe and R. Matsuda, Semistar-operations on integral domains, Math. J. Toyama Univ., 17 (1994), 1–21.
- G. Picozza and F. Tartarone, When the semistar operation ˜? is the identity, Comm. Algebra, 36 (2008), 1954–1975. Mi Hee Park
- Department of Mathematics Chung-Ang University Seoul 156-756, Korea e-mail: mhpark@cau.ac.kr Francesca Tartarone Dipartimento di Matematica Universit`a degli studi Roma Tre Largo San Leonardo Murialdo , 00146 Roma, Italy e-mail: tfrance@mat.uniroma3.it
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI