## Journal of Commutative Algebra

### On finitely stable domains, I

#### Abstract

We prove that an integral domain $R$ is stable and one-dimensional if and only if $R$ is finitely stable and Mori. If $R$ satisfies these two equivalent conditions, then each overring of $R$ also satisfies these conditions, and it is $2$-$v$-generated. We also prove that, if $R$ is an Archimedean stable domain such that $R'$ is local, then $R$ is one-dimensional and so Mori.

#### Article information

Source
J. Commut. Algebra, Volume 11, Number 1 (2019), 49-67.

Dates
First available in Project Euclid: 13 March 2019

https://projecteuclid.org/euclid.jca/1552464132

Digital Object Identifier
doi:10.1216/JCA-2019-11-1-49

Mathematical Reviews number (MathSciNet)
MR3922425

Zentralblatt MATH identifier
07037588

#### Citation

Gabelli, Stefania; Roitman, Moshe. On finitely stable domains, I. J. Commut. Algebra 11 (2019), no. 1, 49--67. doi:10.1216/JCA-2019-11-1-49. https://projecteuclid.org/euclid.jca/1552464132

#### References

• D.D. Anderson, J.A. Huckaba and I.J. Papick, A note on stable domains, Houston J. Math. 13 (1987), 13–17.
• V. Barucci, Mori domains, in Non-Noetherian commutative ring theory; Recent advances, Kluwer Academic Publishers, Dordrecht, 2000.
• H. Bass, On the ubiquity of Gorenstein rings, Math Z. 82 (1963), 8–28.
• S. Bazzoni and L. Salce, Warfield domains, J. Algebra 185 (1996), 836–868.
• A. Bouvier, The local class group of a Krull domain, Canad. Math. Bull. 26 (1983), 13–19.
• M. Fontana, J.A. Huckaba and I.J. Papick, Prüfer domains, Mono. Text. Pure Appl. Math. 203 (1997).
• S. Gabelli and G. Picozza, Star stability and star regularity for Mori domains, Rend. Sem. Mat. Univ. Padova 126 (2011), 107–125.
• S. Gabelli and M. Roitman, On finitely stable domains, II, J. Commutative Algebra, to appear.
• R. Gilmer, Domains in which valuation ideals are prime powers, Arch. Math. 17 (1966), 210–215.
• ––––, Multiplicative ideal theory, Dekker, New York, 1972.
• J. Lipman, Stable ideals and Arf rings J. Pure Appl. Alg. 4 (1974), 319–336.
• J.L. Mott and M. Zafrullah, On Krull domains, Arch. Math. 56 (1991), 559–568.
• J. Ohm, Some counterexamples related to integral closure in $D\lbrack\lbrack x\rbrack\rbrack$, Trans. Amer. Math. Soc. 122 (1966), 321–333.
• B. Olberding, Globalizing local properties of Prüfer domains, J. Algebra 205 (1998), 480–504.
• ––––, Stability, duality and $2$-generated ideals, and a canonical decomposition of modules, Rend. Sem. Mat. Univ. Padova 106 (2001), 261–290.
• ––––, On the classification of stable domains, J. Algebra 243 (2001), 177–197.
• ––––, On the structure of stable domains, Comm. Algebra 30 (2002), 877–895.
• ––––, Noetherian rings without finite normalizations, Progr. Commut. Alg, 2 (2012), 171–203.
• ––––, One-dimensional bad Noetherian domains, Trans. Amer. Math. Soc. 366 (2014), 4067–4095.
• ––––, Finitely stable rings, in Commutative algebra, Recent advances in commutative rings, integer-valued polynomials, and polynomial functions, Springer, New York, 2014.
• B. Olberding, One-dimensional stable rings, J. Algebra 456 (2016), 93–122.
• D.E. Rush, Rings with two-generated ideals, J. Pure Appl. Alg. 73 (1991), 257–275.
• ––––, Two-generated ideals and representations of abelian groups over valuation rings, J. Algebra 177 (1995), 77–101.
• J.D. Sally and W.V. 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.