DOI QR코드

DOI QR Code

GRADED INTEGRAL DOMAINS IN WHICH EACH NONZERO HOMOGENEOUS IDEAL IS DIVISORIAL

  • Chang, Gyu Whan (Department of Mathematics Education Incheon National University) ;
  • Hamdi, Haleh (Department of Pure Mathematics Faculty of Mathematical Sciences University of Tabriz) ;
  • Sahandi, Parviz (Department of Pure Mathematics Faculty of Mathematical Sciences University of Tabriz)
  • Received : 2018.09.15
  • Accepted : 2019.03.04
  • Published : 2019.07.31

Abstract

Let ${\Gamma}$ be a nonzero commutative cancellative monoid (written additively), $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}$ $R_{\alpha}$ be a ${\Gamma}$-graded integral domain with $R_{\alpha}{\neq}\{0\}$ for all ${\alpha}{\in}{\Gamma}$, and $S(H)=\{f{\in}R{\mid}C(f)=R\}$. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if R is integrally closed, then R is a homogeneously divisorial domain if and only if $R_{S(H)}$ is an h-local $Pr{\ddot{u}}fer$ domain whose maximal ideals are invertible, if and only if R satisfies the following four conditions: (i) R is a graded-$Pr{\ddot{u}}fer$ domain, (ii) every homogeneous maximal ideal of R is invertible, (iii) each nonzero homogeneous prime ideal of R is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of R has only finitely many minimal prime ideals. We also show that if R is a graded-Noetherian domain, then R is a homogeneously divisorial domain if and only if $R_{S(H)}$ is a divisorial domain of (Krull) dimension one.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. D. D. Anderson and D. F. Anderson, Divisibility properties of graded domains, Canad. J. Math. 34 (1982), no. 1, 196-215. https://doi.org/10.4153/CJM-1982-013-3 https://doi.org/10.4153/CJM-1982-013-3
  2. D. F. Anderson and G. W. Chang, Graded integral domains and Nagata rings, J. Algebra 387 (2013), 169-184. https://doi.org/10.1016/j.jalgebra.2013.04.021 https://doi.org/10.1016/j.jalgebra.2013.04.021
  3. D. F. Anderson and G. W. Chang, Graded integral domains whose nonzero homogeneous ideals are invertible, Internat. J. Algebra Comput. 26 (2016), no. 7, 1361-1368. https://doi.org/10.1142/S0218196716500582 https://doi.org/10.1142/S0218196716500582
  4. D. F. Anderson, G. W. Chang, and M. Zafrullah, Graded Prufer domains, Comm. Algebra 46 (2018), no. 2, 792-809. https://doi.org/10.1080/00927872.2017.1327595 https://doi.org/10.1080/00927872.2017.1327595
  5. S. Bazzoni and L. Salce, Warfield domains, J. Algebra 185 (1996), no. 3, 836-868. https://doi.org/10.1006/jabr.1996.0353 https://doi.org/10.1006/jabr.1996.0353
  6. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972.
  7. W. Heinzer, Integral domains in which each non-zero ideal is divisorial, Mathematika 15 (1968), 164-170. https://doi.org/10.1112/S0025579300002527 https://doi.org/10.1112/S0025579300002527
  8. B. G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$, J. Algebra 123 (1989), no. 1, 151-170. https://doi.org/10.1016/0021-8693(89)90040-9 https://doi.org/10.1016/0021-8693(89)90040-9
  9. M. Kreuzer and L. Robbiano, Computational Commutative Algebra. 2, Springer-Verlag, Berlin, 2005.
  10. E. Matlis, Cotorsion modules, Mem. Amer. Math. Soc. No. 49 (1964), 66 pp.
  11. E. Matlis, Reflexive domains, J. Algebra 8 (1968), 1-33. https://doi.org/10.1016/0021-8693(68)90031-8 https://doi.org/10.1016/0021-8693(68)90031-8
  12. D. G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge University Press, London, 1968.
  13. C. H. Park and M. H. Park, Integral closure of a graded Noetherian domain, J. Korean Math. Soc. 48 (2011), no. 3, 449-464. https://doi.org/10.4134/JKMS.2011.48.3.449 https://doi.org/10.4134/JKMS.2011.48.3.449
  14. M. H. Park, Integral closure of graded integral domains, Comm. Algebra 35 (2007), no. 12, 3965-3978. https://doi.org/10.1080/00927870701509511 https://doi.org/10.1080/00927870701509511
  15. D. E. Rush, Noetherian properties in monoid rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 259-278. https://doi.org/10.1016/S0022-4049(03)00103-8 https://doi.org/10.1016/S0022-4049(03)00103-8
  16. P. Sahandi, Characterizations of graded Prufer $\star$-multiplication domains, Korean J. Math. 22 (2014), 181-206. https://doi.org/10.11568/kjm.2014.22.1.181
  17. P. Sahandi, Characterizations of graded Prufer $\star$-mulitiplication domains. II, Bull. Iranian Math. Soc. 44 (2018), no. 1, 61-78. https://doi.org/10.1007/s41980-018-0005-1 https://doi.org/10.1007/s41980-018-0005-1
  18. M. Zafrullah, On finite conductor domains, Manuscripta Math. 24 (1978), no. 2, 191-204. https://doi.org/10.1007/BF01310053 https://doi.org/10.1007/BF01310053