• Gmiza, Wafa (Department of Mathematics Faculty of sciences University of Monastir) ;
  • Hizem, Sana (Department of Mathematics Faculty of sciences University of Monastir)
  • Received : 2018.12.20
  • Accepted : 2019.04.01
  • Published : 2019.11.01


Let ${\star}$ be a semistar operation on the integral domain D. In this paper, we prove that D is a $G-{\tilde{\star}}-GCD$ domain if and only if D[X] is a $G-{\star}_1-GCD$ domain if and only if the Nagata ring of D with respect to the semistar operation ${\tilde{\star}}$, $Na(D,{\star}_f)$ is a G-GCD domain if and only if $Na(D,{\star}_f)$ is a GCD domain, where ${\star}_1$ is the semistar operation on D[X] introduced by G. Picozza [12].


  1. D. D. Anderson, Multiplication ideals, multiplication rings, and the ring R(X), Canad. J. Math. 28 (1976), no. 4, 760-768.
  2. D. D. Anderson and D. F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Paul. 28 (1980), no. 2, 215-221.
  3. G. W. Chang and M. Fontana, Uppers to zero and semistar operations in polynomial rings, J. Algebra 318 (2007), no. 1, 484-493.
  4. S. El-Baghdadi, Semistar GCD domains, Comm. Algebra 38 (2010), no. 8, 3029-3044.
  5. M. Fontana and J. A. Huckaba, Localizing systems and semistar operations, in Non-Noetherian commutative ring theory, 169-197, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000.
  6. M. Fontana and K. A. Loper, Kronecker function rings: a general approach, in Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), 189-205, Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001.
  7. M. Fontana and K. A. Loper, Nagata rings, Kronecker function rings, and related semistar operations, Comm. Algebra 31 (2003), no. 10, 4775-4805.
  8. M. Fontana and G. Picozza, Semistar invertibility on integral domains, Algebra Colloq. 12 (2005), no. 4, 645-664.
  9. W. Gmiza and S. Hizem, Semistar ascending chain conditions over polynomial rings, Submitted.
  10. A. Okabe and R. Matsuda, Semistar-operations on integral domains, Math. J. Toyama Univ. 17 (1994), 1-21.
  11. B. Olberding, Characterizations and constructions of h-local domains, in Models, modules and abelian groups, 385-406, Walter de Gruyter, Berlin, 2008.
  12. G. Picozza, A note on semistar Noetherian domains, Houston J. Math. 33 (2007), no. 2, 415-432.
  13. J. Querre, Sur le groupe des classes de diviseurs, C. R. Acad. Sci. Paris Ser. A-B 284 (1977), no. 7, A397-A399.
  14. D. Spirito, Jaffard families and localizations of star operations, to appear in J. Commut. Algebra.