# Normal Pairs of Going-down Rings

Dobbs, David Earl;Shapiro, Jay Allen

• Received : 2010.06.04
• Accepted : 2010.12.09
• Published : 2011.03.31
• 26 17

#### Abstract

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.

#### Keywords

Normal pair;prime ideal;total quotient ring;valuation domain;divided domain;pullback;going-down ring;EGD ring;locally divided ring;weak Baer ring;reduced ring

#### References

1. A. Badawi, On divided commutative rings, Comm. Algebra, 27(1999), 1465-1474. https://doi.org/10.1080/00927879908826507
2. A. Badawi and D. E. Dobbs, On locally divided rings and going-down rings, Comm. Algebra, 29(2001), 2805-2825. https://doi.org/10.1081/AGB-100104988
3. J. Coykendall and D. E. Dobbs, Saturated chains of integrally closed overrings, JP J. Algebra, Number Theory Appl., 13(2009), 121-130.
4. E. D. Davis, Overrings of commutative rings. III: normal pairs, Trans. Amer. Math. Soc., 182(1973), 175-185.
5. D. E. Dobbs, On going-down for simple overrings, II, Comm. Algebra, 1(1974), 439-458. https://doi.org/10.1080/00927877408548715
6. D. E. Dobbs, Divided rings and going-down, Pac. J. Math., 67(1976), 353-363. https://doi.org/10.2140/pjm.1976.67.353
7. D. E. Dobbs, On Henselian pullbacks, pp. 317-326, in Lecture Notes Pure Appl. Math. 189, Dekker, New York, 1997.
8. D. E. Dobbs, Going-down rings with zero-divisors, Houston J. Math., 23(1997), 1-12.
9. D. E. Dobbs, When is a pullback a locally divided domain?, Houston J. Math., 35(2009), 341-351.
10. D. E. Dobbs, When a minimal overring is a going-down domain, Houston J. Math., 36(2010), 33-42.
11. D. E. Dobbs and I. J. Papick, On going-down for simple overrings, III, Proc. Amer. Math. Soc., 54(1976), 147-168.
12. D. E. Dobbs and G. Picavet, Weak Baer going-down rings, Houston J. Math., 29(2003), 559-581.
13. D. E. Dobbs and J. Shapiro, A generalization of divided domains and its connection to weak Baer going-down rings, Comm. Algebra, 37(2009), 3553-3572. https://doi.org/10.1080/00927870902828488
14. D. E. Dobbs and J. Shapiro, INC-extensions amid zero-divisors, Int. Electron. J. Algebra, 7(2010), 102-109.
15. D. E. Dobbs and J. Shapiro, Normal pairs with zero-divisors, J. Algebra Appl., to appear.
16. D. Ferrand and J.-P. Olivier, Morphismes minimaux, J. Algebra, 16(1970), 461-471. https://doi.org/10.1016/0021-8693(70)90020-7
17. M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl., 123(1980), 331-355. https://doi.org/10.1007/BF01796550
18. R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
19. M. Griffin, Prufer rings with zero divisors, J. Reine Angew. Math., 240(1970), 55-67.
20. J. A. Huckaba, Commutative Rings with Zero Divisors, Dekker, New York, 1988.
21. I. Kaplansky, Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, 1974.
22. S. McAdam, Simple going down, J. London Math. Soc., 13(1976), 167-173. https://doi.org/10.1112/jlms/s2-13.1.167
23. M. Nagata, Local Rings, Wiley-Interscience, New York, 1962.
24. J. Sato, T. Sugatani and K. I. Yoshida, On minimal overrings of a Noetherian domain, Comm. Algebra, 20(1992), 1746-1753.

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2. On Finite Maximal Chains of Weak Baer Going-Down Rings vol.40, pp.5, 2012, https://doi.org/10.1080/00927872.2011.558881
3. A GENERALIZATION OF PRÜFER'S ASCENT RESULT TO NORMAL PAIRS OF COMPLEMENTED RINGS vol.10, pp.06, 2011, https://doi.org/10.1142/S021949881100521X