On Strong Going-Between, Going-Down, And Their Universalizations, II
Annales mathématiques Blaise Pascal, Volume 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 $A\subseteq B$ are domains such that $A$ is an LFD universally going-down domain and $B$ is algebraic over $A$, then the inclusion map $A\left[{X}_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{X}_{n}\right]↪B\left[{X}_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{X}_{n}\right]$ satisfies GB for each $n\ge 0$. However, for any nonzero ring $A$ and indeterminate $X$ over $A$, the inclusion map $A↪A\left[X\right]$ is not universally (S)GB.

DOI: 10.5802/ambp.175
David E. Dobbs 1; Gabriel Picavet 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 = {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/}
}
TY  - JOUR
AU  - David E. Dobbs
AU  - Gabriel Picavet
TI  - On Strong Going-Between, Going-Down, And Their Universalizations, II
JO  - Annales mathématiques Blaise Pascal
PY  - 2003
DA  - 2003///
SP  - 245
EP  - 260
VL  - 10
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.175/
UR  - https://www.ams.org/mathscinet-getitem?mr=2031270
UR  - https://zbmath.org/?q=an%3A1071.13003
UR  - https://doi.org/10.5802/ambp.175
DO  - 10.5802/ambp.175
LA  - en
ID  - AMBP_2003__10_2_245_0
ER  - 
%0 Journal Article
%A David E. Dobbs
%A Gabriel Picavet
%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 https://doi.org/10.5802/ambp.175
%R 10.5802/ambp.175
%G en
%F AMBP_2003__10_2_245_0
David E. Dobbs; Gabriel Picavet. On Strong Going-Between, Going-Down, And Their Universalizations, II. Annales mathématiques Blaise Pascal, Volume 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 | DOI | MR | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

Cited by Sources: