BibTex RIS Kaynak Göster

CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS

Yıl 2014, , 99 - 114, 01.12.2014
https://doi.org/10.24330/ieja.266230

Öz

Let R be a (commutative integral) domain with quotient K; let
R0 be the integral closure of R (in K). Then each overring of R (inside K) is a
going-down domain if and only if R0 is a locally pseudo-valuation domain, T ⊆T0
satisfies going-down for every overring T of R, and tr. deg[VR0 (M)/M(R0)M :R0/M] ≤ 1 for every maximal ideal M of R0
(where VR0 (M) denotes the
valuation domain that is canonically associated to the pseudo-valuation domain
(R0)M). Additional equivalences are given in case R is locally finitedimensional.
Applications include the case where R is integrally closed or R
is not a Jaffard domain or R[X] is catenarian.

Kaynakça

  • D. D. Anderson and D. E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math., 32 (1980), 362–384.
  • D. F. Anderson, A. Bouvier, D. E. Dobbs, M. Fontana and S. Kabbaj, On Jaffard domains, Exposition. Math., 6 (1988), 145–175.
  • A. Ayache, Integrally closed domains with treed overrings, Ric. Mat. 63(1) (2014), 93-100.
  • A. Ayache, M. Ben Nasr, O. Echi and N. Jarboui, Universally catenarian and going-down pairs of rings, Math. Z., 238 (2001), 695–731.
  • A. Ayache and P. J. Cahen, Anneaux v´erifiant absolument l’in´egalit´e ou la formule de la dimension, Boll. Un. Mat. Ital. B (7), 6 (1992), 39–65.
  • A. Ayache and N. Jarboui, An answer to a Dobbs conjecture about treed domains, J. Algebra, 320 (2008), 3720–3725.
  • A. Ayache, N. Jarboui and E. Massaoud, Pairs of domains where all inter- mediates domains are treed, Arab J. Sci. Eng., 36 (2011), 933–946.
  • E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M , Michigan Math. J., 20 (1973), 79–95.
  • M. Ben Nasr and N. Jarboui, Maximal non-Jaffard subrings of a field, Pub. Mat., 44 (2000), 157–175.
  • A. Bouvier, D. E. Dobbs and M. Fontana, Universally catenarian integral domains, Adv. Math., 72 (1988), 211–238.
  • A. Bouvier, D. E. Dobbs and M. Fontana, Two sufficient conditions for universal catenarity, Comm. Algebra, 15 (1987), 861–872.
  • R. D. Chatham and D. E. Dobbs, On pseudo-valuation domains whose over- rings are going-down domains, Houston J. Math., 28 (2002), 13–19.
  • D. E. Dobbs, On going-down for simple overrings, II, Comm. Algebra, 1 (1974), 439–458.
  • D. E. Dobbs, Ascent and descent of going-down rings for integral extensions, Bull. Austral. Math. Soc., 15 (1976), 253–264.
  • D. E. Dobbs, Divided rings and going-down, Pacific J. Math., 67 (1976), –363.
  • D. E. Dobbs, Coherence, ascent of going-down and pseudo-valuation do- mains, Houston J. Math., 4 (1978), 551–567.
  • D. E. Dobbs, On treed overrings and going-down domains, Rend. Mat., 7 (1987), 317–322.
  • D. E. Dobbs and M. Fontana, Locally pseudo-valuation domains, Ann. Mat. Pura Appl., 134(4) (1983), 147–168.
  • D. E. Dobbs and M. Fontana, Integral overrings of two-dimensional going- down domains, Proc. Amer. Math. Soc., 115(3) (1992), 655–662.
  • D. E. Dobbs, M. Fontana and I. J. Papick, Direct limits and going-down, Comment. Math. Univ. St. Paul., 31(2) (1982), 129–135.
  • D. E. Dobbs and I. J. Papick, On going-down for simple overrings, III, Proc. Amer. Math. Soc., 54 (1976), 35–38.
  • D. E. Dobbs and I. J. Papick, Going-down, a survey, Nieuw Arch. v. Wisk, (1978), 255–291.
  • M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl., 123 (1980), 331–355.
  • M. Fontana, Carr´es cart´esiens et anneaux de pseudo-valuation, Pub. Math. Univ. Lyon, 17 (1980), 57–95.
  • M. Fontana and E. Houston, On integral domains whose overrings are Ka- plansky ideal transforms, J. Pure Appl. Algebra, 163 (2001), 173–192.
  • R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
  • R. Gilmer and W. J. Heinzer, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ., 7 (1967), 133–150.
  • J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, Pacific J. Math., 75 (1978), 137–147.
  • J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, II, Houston. J. Math., 4 (1978), 199–207.
  • S. Malik and J. L. Mott, Strong S-domains, J. Pure Appl. Algebra, 28 (1983), 249–264.
  • S. McAdam, Simple going-down, J. London Math. Soc., 13 (1976), 167–173.
  • S. McAdam, Going down: ascent/descent, Comm. Algebra, 29 (2001), 5191–
  • I. J. Papick, Topologically defined classes of going-down domains, Trans. Amer. Math. Soc., 219 (1976), 1–37. Ahmed Ayache Faculty of Science Sana’a University
  • P.O. Box 12460, Sana’a, Yemen e-mail: aaayache@yahoo.com David E. Dobbs
  • Department of Mathematics University of Tennessee Knoxville, TN 37996-1320, U.S.A. e-mail: ddobbs1@utk.edu
Yıl 2014, , 99 - 114, 01.12.2014
https://doi.org/10.24330/ieja.266230

Öz

Kaynakça

  • D. D. Anderson and D. E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math., 32 (1980), 362–384.
  • D. F. Anderson, A. Bouvier, D. E. Dobbs, M. Fontana and S. Kabbaj, On Jaffard domains, Exposition. Math., 6 (1988), 145–175.
  • A. Ayache, Integrally closed domains with treed overrings, Ric. Mat. 63(1) (2014), 93-100.
  • A. Ayache, M. Ben Nasr, O. Echi and N. Jarboui, Universally catenarian and going-down pairs of rings, Math. Z., 238 (2001), 695–731.
  • A. Ayache and P. J. Cahen, Anneaux v´erifiant absolument l’in´egalit´e ou la formule de la dimension, Boll. Un. Mat. Ital. B (7), 6 (1992), 39–65.
  • A. Ayache and N. Jarboui, An answer to a Dobbs conjecture about treed domains, J. Algebra, 320 (2008), 3720–3725.
  • A. Ayache, N. Jarboui and E. Massaoud, Pairs of domains where all inter- mediates domains are treed, Arab J. Sci. Eng., 36 (2011), 933–946.
  • E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M , Michigan Math. J., 20 (1973), 79–95.
  • M. Ben Nasr and N. Jarboui, Maximal non-Jaffard subrings of a field, Pub. Mat., 44 (2000), 157–175.
  • A. Bouvier, D. E. Dobbs and M. Fontana, Universally catenarian integral domains, Adv. Math., 72 (1988), 211–238.
  • A. Bouvier, D. E. Dobbs and M. Fontana, Two sufficient conditions for universal catenarity, Comm. Algebra, 15 (1987), 861–872.
  • R. D. Chatham and D. E. Dobbs, On pseudo-valuation domains whose over- rings are going-down domains, Houston J. Math., 28 (2002), 13–19.
  • D. E. Dobbs, On going-down for simple overrings, II, Comm. Algebra, 1 (1974), 439–458.
  • D. E. Dobbs, Ascent and descent of going-down rings for integral extensions, Bull. Austral. Math. Soc., 15 (1976), 253–264.
  • D. E. Dobbs, Divided rings and going-down, Pacific J. Math., 67 (1976), –363.
  • D. E. Dobbs, Coherence, ascent of going-down and pseudo-valuation do- mains, Houston J. Math., 4 (1978), 551–567.
  • D. E. Dobbs, On treed overrings and going-down domains, Rend. Mat., 7 (1987), 317–322.
  • D. E. Dobbs and M. Fontana, Locally pseudo-valuation domains, Ann. Mat. Pura Appl., 134(4) (1983), 147–168.
  • D. E. Dobbs and M. Fontana, Integral overrings of two-dimensional going- down domains, Proc. Amer. Math. Soc., 115(3) (1992), 655–662.
  • D. E. Dobbs, M. Fontana and I. J. Papick, Direct limits and going-down, Comment. Math. Univ. St. Paul., 31(2) (1982), 129–135.
  • D. E. Dobbs and I. J. Papick, On going-down for simple overrings, III, Proc. Amer. Math. Soc., 54 (1976), 35–38.
  • D. E. Dobbs and I. J. Papick, Going-down, a survey, Nieuw Arch. v. Wisk, (1978), 255–291.
  • M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl., 123 (1980), 331–355.
  • M. Fontana, Carr´es cart´esiens et anneaux de pseudo-valuation, Pub. Math. Univ. Lyon, 17 (1980), 57–95.
  • M. Fontana and E. Houston, On integral domains whose overrings are Ka- plansky ideal transforms, J. Pure Appl. Algebra, 163 (2001), 173–192.
  • R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
  • R. Gilmer and W. J. Heinzer, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ., 7 (1967), 133–150.
  • J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, Pacific J. Math., 75 (1978), 137–147.
  • J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, II, Houston. J. Math., 4 (1978), 199–207.
  • S. Malik and J. L. Mott, Strong S-domains, J. Pure Appl. Algebra, 28 (1983), 249–264.
  • S. McAdam, Simple going-down, J. London Math. Soc., 13 (1976), 167–173.
  • S. McAdam, Going down: ascent/descent, Comm. Algebra, 29 (2001), 5191–
  • I. J. Papick, Topologically defined classes of going-down domains, Trans. Amer. Math. Soc., 219 (1976), 1–37. Ahmed Ayache Faculty of Science Sana’a University
  • P.O. Box 12460, Sana’a, Yemen e-mail: aaayache@yahoo.com David E. Dobbs
  • Department of Mathematics University of Tennessee Knoxville, TN 37996-1320, U.S.A. e-mail: ddobbs1@utk.edu
Toplam 35 adet kaynakça vardır.

Ayrıntılar

Diğer ID JA73MB53BC
Bölüm Makaleler
Yazarlar

Ahmed Ayache Bu kişi benim

David E. Dobbs Bu kişi benim

Yayımlanma Tarihi 1 Aralık 2014
Yayımlandığı Sayı Yıl 2014

Kaynak Göster

APA Ayache, A., & Dobbs, D. E. (2014). CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS. International Electronic Journal of Algebra, 16(16), 99-114. https://doi.org/10.24330/ieja.266230
AMA Ayache A, Dobbs DE. CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS. IEJA. Aralık 2014;16(16):99-114. doi:10.24330/ieja.266230
Chicago Ayache, Ahmed, ve David E. Dobbs. “CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS”. International Electronic Journal of Algebra 16, sy. 16 (Aralık 2014): 99-114. https://doi.org/10.24330/ieja.266230.
EndNote Ayache A, Dobbs DE (01 Aralık 2014) CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS. International Electronic Journal of Algebra 16 16 99–114.
IEEE A. Ayache ve D. E. Dobbs, “CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS”, IEJA, c. 16, sy. 16, ss. 99–114, 2014, doi: 10.24330/ieja.266230.
ISNAD Ayache, Ahmed - Dobbs, David E. “CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS”. International Electronic Journal of Algebra 16/16 (Aralık 2014), 99-114. https://doi.org/10.24330/ieja.266230.
JAMA Ayache A, Dobbs DE. CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS. IEJA. 2014;16:99–114.
MLA Ayache, Ahmed ve David E. Dobbs. “CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS”. International Electronic Journal of Algebra, c. 16, sy. 16, 2014, ss. 99-114, doi:10.24330/ieja.266230.
Vancouver Ayache A, Dobbs DE. CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS. IEJA. 2014;16(16):99-114.