Journal of Commutative Algebra

Factoring ideals and stability in integral domains

A. Mimouni

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In an integral domain $R$, a nonzero ideal is called a \textit {weakly $ES$-stable ideal} if it can be factored into a product of an invertible ideal and an idempotent ideal of $R$; and $R$ is called a \textit {weakly $ES$-stable domain} if every nonzero ideal is a weakly $ES$-stable ideal. This paper studies the notion of weakly $ES$-stability in various contexts of integral domains such as Noetherian and Mori domains, valuation and Pr\"ufer domains, pullbacks and more. In particular, we establish strong connections between this notion and well-known stability conditions, namely, Lipman, Sally-Vasconcelos and Eakin-Sathaye stabilities.

Article information

Source
J. Commut. Algebra, Volume 9, Number 2 (2017), 263-290.

Dates
First available in Project Euclid: 3 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.jca/1496476824

Digital Object Identifier
doi:10.1216/JCA-2017-9-2-263

Mathematical Reviews number (MathSciNet)
MR3659951

Zentralblatt MATH identifier
1370.13004

Subjects
Primary: 13A15: Ideals; multiplicative ideal theory 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13G05: Integral domains
Secondary: 13F30: Valuation rings [See also 13A18] 13G05: Integral domains

Keywords
Invertible ideal idempotent ideal weakly $ES$-stable domains weakly $ES$-stable ideals strongly stable ideal valuation domain Prüfer domain Noetherian domain pullbacks

Citation

Mimouni, A. Factoring ideals and stability in integral domains. J. Commut. Algebra 9 (2017), no. 2, 263--290. doi:10.1216/JCA-2017-9-2-263. https://projecteuclid.org/euclid.jca/1496476824


Export citation

References

  • D.D. Anderson, J. Huckaba and I. Papick, A note on stable domains, Houston J. Math. 13 (1987), 13–17.
  • D.D. Anderson and B.G. Kang, Pseudo-Dedekind domains and divisorial ideals in $R[X]_{T}$, J. Algebra 122 (1989), 323-–336.
  • D.F. Anderson, V. Barucci and D. Dobbs, Coherent Mori domains and the principal ideal theorem, Comm. Algebra 15 (1987), 1119-–1156.
  • D.F. Anderson and D.E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math. 32 (1980), 362–384.
  • V. Barucci, Mori domains, Non-Noetherian commutative ring theory, Kluwer, Dordrecht, 2000.
  • V. Barucci and E. Houston, On the prime spectrum of a Mori domain, Comm. Algebra 24 (1996), 3599–3622.
  • E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form $D+M$, Michigan Math. J. 20 (1992), 79–95.
  • S. Bazzoni, Class semigroup of Prüfer domains, J. Algebra 184 (1996), 613–631.
  • ––––, Clifford regular domains, J. Algebra 238 (2001), 703–722.
  • ––––, Finite character of finitely stable domains, J. Pure Appl. Algebra 215 (2011), 1127-–1132.
  • S. Bazzoni and L. Salce, Warfield domains, J. Algebra 185 (1996), 836–-868.
  • D.E. Dobbs and R. Fedder, Conducive integral domains, J. Algebra 86 (1984), 494–510.
  • P. Eakin and A. Sathaye, Prestable ideals, J. Algebra 41 (1976), 439–454.
  • W. Fangui and R.L. McCasland, On strong Mori domains, J. Pure Appl. Algebra 135 (1999), 155–-165.
  • M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl. 123 (1980), 331–355.
  • M. Fontana and S. Gabelli, On the class group and local class group of a pullback, J. Algebra 181 (1996) 803–835.
  • M. Fontana, J. Huckaba and I. Papick, Prüfer domains, Mono. Text. Pure Appl. Math. 203, Marcel Dekker, Inc., New York, 1997.
  • M. Fontana, J.A. Huckaba and I.J. Papick, Domains satisfying the trace property, J. Algebra 107 (1987), 169–182.
  • S. Gabelli, Ten problems on stability of domains, in Commutative algebra, Fontana, Frish and Galz, eds., Springer, Berlin, 2014.
  • S. Gabelli and E. Houston, Coherentlike conditions in pullbacks, Michigan Math. J. 44 (1997) 99–122.
  • S. Gabelli and M. Roitman, On finitely stable domains, submitted.
  • R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972.
  • J. Hedstrom and E. Houston, Pseudo-valuation domains, Pacific J. Math. 75 (1978), 137–147.
  • S. Kabbaj and A. Mimouni, Class semigroups of integral domains, J. Algebra 264 (2003), 620–-640
  • I. Kaplansky, Commutative rings, revised edition, Chicago University Press, Chicago, 1972.
  • J. Lipman, Stable ideals and Arf rings, American J. Math. 93 (1971), 649–685.
  • B. Olberding, On the classification of stable domains, J. Algebra 243 (2001), 177–197.
  • ––––, On the structure of stable domains, Comm. Algebra 30 (2002), 877–895.
  • ––––, Duality, stability, $2$-generated ideals and a decompostion of modules, Rend. Sem. Mat. 106 (2001), 261–-290.
  • ––––, Stability of ideals and its applications, Ideal theoretic methods in commutative algebra, Lect. Notes Pure Appl. Math 220, Dekker, New York, 2010.
  • ––––, One-dimensional bad Noetherian domains, Trans. Amer. Math. Soc. 366 (2014), 4067–4095.
  • J. Querre, Sur une propriete des anneaux de Krull, Bull. Sci. Math. 95 (1971), 341–354.
  • J. Sally and W. Vasconcelos, Stable rings and a problem of Bass, Bull. Amer. Math. Soc. 79 (1973), 574–576.
  • ––––, Stable rings, J. Pure Appl. Alg. 4 (1974), 319–336.