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

#### Ahmed Ayache [1] , David E. Dobbs [2]

##### 163 210

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.
Going-down domain, overring, treed domain, pseudo-valuation domain, valuation domain
• 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
Konular JA73MB53BC Makaleler Yazar: Ahmed Ayache Yazar: David E. Dobbs
 Bibtex @ { ieja266230, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2014}, volume = {16}, pages = {99 - 114}, doi = {10.24330/ieja.266230}, title = {CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS}, key = {cite}, author = {Dobbs, David E. and Ayache, Ahmed} } APA Ayache, A , Dobbs, D . (2014). CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS. International Electronic Journal of Algebra, 16 (16), 99-114. DOI: 10.24330/ieja.266230 MLA Ayache, A , Dobbs, D . "CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS". International Electronic Journal of Algebra 16 (2014): 99-114 Chicago Ayache, A , Dobbs, D . "CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS". International Electronic Journal of Algebra 16 (2014): 99-114 RIS TY - JOUR T1 - CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS AU - Ahmed Ayache , David E. Dobbs Y1 - 2014 PY - 2014 N1 - doi: 10.24330/ieja.266230 DO - 10.24330/ieja.266230 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 99 EP - 114 VL - 16 IS - 16 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.266230 UR - http://dx.doi.org/10.24330/ieja.266230 Y2 - 2019 ER - EndNote %0 International Electronic Journal of Algebra CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS %A Ahmed Ayache , David E. Dobbs %T CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS %D 2014 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 16 %N 16 %R doi: 10.24330/ieja.266230 %U 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. http://dx.doi.org/10.24330/ieja.266230