On Strong Going-Between, Going-Down, And Their Universalizations, II
Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 245-260.

We consider analogies between the logically independent properties of strong going-between (SGB) and going-down (GD), as well as analogies between the universalizations of these properties. Transfer results are obtained for the (universally) SGB property relative to pullbacks and Nagata ring constructions. It is shown that if AB are domains such that A is an LFD universally going-down domain and B is algebraic over A, then the inclusion map A[X 1 ,,X n ]B[X 1 ,,X n ] satisfies GB for each n0. However, for any nonzero ring A and indeterminate X over A, the inclusion map AA[X] is not universally (S)GB.

DOI : 10.5802/ambp.175
Dobbs, David E. 1 ; Picavet, Gabriel 2

1 University of Tennessee Department of Mathematics Knoxville, Tennessee 37996-1300 U.S.A.
2 Université Blaise Pascal Laboratoire de Mathématiques Pures 63177 Aubière Cedex FRANCE
@article{AMBP_2003__10_2_245_0,
     author = {Dobbs, David E. and Picavet, Gabriel},
     title = {On {Strong} {Going-Between,} {Going-Down,} {And} {Their} {Universalizations,} {II}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {245--260},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {10},
     number = {2},
     year = {2003},
     doi = {10.5802/ambp.175},
     zbl = {1071.13003},
     mrnumber = {2031270},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/ambp.175/}
}
TY  - JOUR
AU  - Dobbs, David E.
AU  - Picavet, Gabriel
TI  - On Strong Going-Between, Going-Down, And Their Universalizations, II
JO  - Annales mathématiques Blaise Pascal
PY  - 2003
SP  - 245
EP  - 260
VL  - 10
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - http://archive.numdam.org/articles/10.5802/ambp.175/
DO  - 10.5802/ambp.175
LA  - en
ID  - AMBP_2003__10_2_245_0
ER  - 
%0 Journal Article
%A Dobbs, David E.
%A Picavet, Gabriel
%T On Strong Going-Between, Going-Down, And Their Universalizations, II
%J Annales mathématiques Blaise Pascal
%D 2003
%P 245-260
%V 10
%N 2
%I Annales mathématiques Blaise Pascal
%U http://archive.numdam.org/articles/10.5802/ambp.175/
%R 10.5802/ambp.175
%G en
%F AMBP_2003__10_2_245_0
Dobbs, David E.; Picavet, Gabriel. On Strong Going-Between, Going-Down, And Their Universalizations, II. Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 245-260. doi : 10.5802/ambp.175. http://archive.numdam.org/articles/10.5802/ambp.175/

[1] Anderson, D. F.; Dobbs, D. E.; Fontana, M. On treed Nagata rings, J. Pure Appl. Algebra, Volume 61 (1989), pp. 107-122 | DOI | MR | Zbl

[2] Bourbaki, N. Commutative Algebra, Addison-Wesley, Reading, 1972 | Zbl

[3] Bouvier, A.; Dobbs, D. E.; Fontana, M. Universally catenarian integral domains, Adv. in Math., Volume 72 (1988), pp. 211-238 | DOI | MR | Zbl

[4] Dobbs, D. E. On going-down for simple overrings, II, Comm. Algebra, Volume 1 (1974), pp. 439-458 | DOI | MR | Zbl

[5] Dobbs, D. E. Going-down rings with zero-divisors, Houston J. Math., Volume 23 (1997), pp. 1-12 | MR | Zbl

[6] Dobbs, D. E.; Fontana, M. Universally going-down homomorphisms of commutative rings, J. Algebra, Volume 90 (1984), pp. 410-429 | DOI | MR | Zbl

[7] Dobbs, D. E.; Fontana, M. Universally going-down integral domains, Arch. Math., Volume 42 (1984), pp. 426-429 | DOI | MR | Zbl

[8] Dobbs, D. E.; Papick, I. J. On going-down for simple overrings, III, Proc. Amer. Math. Soc., Volume 54 (1976), pp. 35-38 | DOI | MR | Zbl

[9] Dobbs, D. E.; Picavet, G. On strong going-between, going-down, and their universalizations, Rings, Modules, Algebras and Abelian Groups, Dekker, New York, to appear | MR | Zbl

[10] Fontana, M. Topologically defined classes of commutative rings, Ann. Mat. Pura Appl., Volume 123 (1980), pp. 331-355 | DOI | MR | Zbl

[11] Fontana, M.; Huckaba, J. A.; Papick, I. J. Prüfer Domains, Dekker, New York, 1997 | MR | Zbl

[12] Gilmer, R. Multiplicative Ideal Theory, Dekker, New York, 1972 | MR | Zbl

[13] Grothendieck, A.; Dieudonné, J. A. Eléments de Géométrie Algébrique, Springer-Verlag, Berlin, 1971 | Zbl

[14] Hochster, M. Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., Volume 142 (1969), pp. 43-60 | DOI | MR | Zbl

[15] Kaplansky, I. Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, 1974 | Zbl

[16] Lewis, W. J. The spectrum of a ring as a partially ordered set, J. Algebra, Volume 25 (1973), pp. 419-434 | DOI | MR | Zbl

[17] McAdam, S. Going down in polynomial rings, Can. J. Math., Volume 23 (1971), pp. 704-711 | DOI | MR | Zbl

[18] Picavet, G. Universally going-down rings, 1-split rings and absolute integral closure, Comm. Algebra, Volume 31 (2003), pp. 4655-4685 | DOI | MR | Zbl

[19] Ratliff, Jr., L. J. Going-between rings and contractions of saturated chains of prime ideals, Rocky Mountain J. Math., Volume 7 (1977), pp. 777-787 | DOI | MR | Zbl

[20] Ratliff, Jr., L. J. A(X) and GB-Noetherian rings, Rocky Mountain J. Math., Volume 9 (1979), pp. 337-353 | MR | Zbl

Cité par Sources :