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.

@article{AMBP_2003__10_2_245_0,
     author = {David E. Dobbs and Gabriel Picavet},
     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},
     mrnumber = {2031270},
     zbl = {1071.13003},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.175/}
}
David E. Dobbs; Gabriel Picavet. 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.175/

[1] D. F. Anderson; D. E. Dobbs; M. Fontana On treed Nagata rings, J. Pure Appl. Algebra, Volume 61 (1989), pp. 107-122 | Article | MR 1025917 | Zbl 0691.13005

[2] N. Bourbaki Commutative Algebra, Addison-Wesley, Reading, 1972 | Zbl 0279.13001

[3] A. Bouvier; D. E. Dobbs; M. Fontana Universally catenarian integral domains, Adv. in Math., Volume 72 (1988), pp. 211-238 | Article | MR 972761 | Zbl 0695.13014

[4] D. E. Dobbs On going-down for simple overrings, II, Comm. Algebra, Volume 1 (1974), pp. 439-458 | Article | MR 364225 | Zbl 0285.13001

[5] D. E. Dobbs Going-down rings with zero-divisors, Houston J. Math., Volume 23 (1997), pp. 1-12 | MR 1688682 | Zbl 0896.13006

[6] D. E. Dobbs; M. Fontana Universally going-down homomorphisms of commutative rings, J. Algebra, Volume 90 (1984), pp. 410-429 | Article | MR 760019 | Zbl 0544.13004

[7] D. E. Dobbs; M. Fontana Universally going-down integral domains, Arch. Math., Volume 42 (1984), pp. 426-429 | Article | MR 756695 | Zbl 0526.13007

[8] D. E. Dobbs; I. J. Papick On going-down for simple overrings, III, Proc. Amer. Math. Soc., Volume 54 (1976), pp. 35-38 | Article | MR 417153 | Zbl 0285.13002

[9] D. E. Dobbs; G. Picavet On strong going-between, going-down, and their universalizations, Rings, Modules, Algebras and Abelian Groups, Dekker, New York, to appear | MR 2050708 | Zbl 1069.13008

[10] M. Fontana Topologically defined classes of commutative rings, Ann. Mat. Pura Appl., Volume 123 (1980), pp. 331-355 | Article | MR 581935 | Zbl 0443.13001

[11] M. Fontana; J. A. Huckaba; I. J. Papick Prüfer Domains, Dekker, New York, 1997 | MR 1413297 | Zbl 0861.13006

[12] R. Gilmer Multiplicative Ideal Theory, Dekker, New York, 1972 | MR 427289 | Zbl 0248.13001

[13] A. Grothendieck; J. A. Dieudonné Eléments de Géométrie Algébrique, Springer-Verlag, Berlin, 1971 | Zbl 0203.23301

[14] M. Hochster Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., Volume 142 (1969), pp. 43-60 | Article | MR 251026 | Zbl 0184.29401

[15] I. Kaplansky Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, 1974 | Zbl 0296.13001

[16] W. J. Lewis The spectrum of a ring as a partially ordered set, J. Algebra, Volume 25 (1973), pp. 419-434 | Article | MR 314811 | Zbl 0266.13010

[17] S. McAdam Going down in polynomial rings, Can. J. Math., Volume 23 (1971), pp. 704-711 | Article | MR 280482 | Zbl 0223.13006

[18] G. Picavet Universally going-down rings, 1-split rings and absolute integral closure, Comm. Algebra, Volume 31 (2003), pp. 4655-4685 | Article | MR 1998022 | Zbl 01980552

[19] L. J. Ratliff, Jr. Going-between rings and contractions of saturated chains of prime ideals, Rocky Mountain J. Math., Volume 7 (1977), pp. 777-787 | Article | MR 447207 | Zbl 0372.13008

[20] L. J. Ratliff, Jr. A(X) and GB-Noetherian rings, Rocky Mountain J. Math., Volume 9 (1979), pp. 337-353 | MR 519947 | Zbl 0426.13006