Rocky Mountain Journal of Mathematics

The ascending chain condition on principal ideals in composite generalized power series rings

Jung Wook Lim and Dong Yeol Oh

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Abstract

Let $D \subseteq E$ be an extension of commutative rings with identity, $I$ a nonzero proper ideal of $D$, $(\Gamma , \leq )$ a strictly totally ordered monoid such that $0 \leq \alpha $ for all $\alpha \in \Gamma $, and $\Gamma ^*=\Gamma \setminus \{0\}$. Let $D+[\![E^{\Gamma ^*, \leq }]\!]=\{f \in [\![E^{\Gamma , \leq }]\!] \mid f(0) \in D\}$ and $D+[\![I^{\Gamma ^*, \leq }]\!] =\{f \in [\![D^{\Gamma , \leq }]\!] \mid f(\alpha ) \in I$ for all $\alpha \in \Gamma ^*\}$. In this paper, we give some conditions for the rings $D+[\![E^{\Gamma ^*, \leq }]\!]$ and $D+[\![I^{\Gamma ^*, \leq }]\!]$ to satisfy the ascending chain condition on principal ideals.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1223-1236.

Dates
First available in Project Euclid: 29 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1567044037

Digital Object Identifier
doi:10.1216/RMJ-2019-49-4-1223

Mathematical Reviews number (MathSciNet)
MR3998919

Zentralblatt MATH identifier
07104715

Subjects
Primary: 13A02: Graded rings [See also 16W50] 13A15: Ideals; multiplicative ideal theory 13E99: None of the above, but in this section 13G05: Integral domains

Keywords
Generalized power series rings ring extensions ascending chain condition on principal ideals

Citation

Lim, Jung Wook; Oh, Dong Yeol. The ascending chain condition on principal ideals in composite generalized power series rings. Rocky Mountain J. Math. 49 (2019), no. 4, 1223--1236. doi:10.1216/RMJ-2019-49-4-1223. https://projecteuclid.org/euclid.rmjm/1567044037


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References

  • D.D. Anderson and M. Winders, Idealization of a module, J. Commutative Algebra 1 (2009), 3–56.
    Zentralblatt MATH: 1194.13002
    Digital Object Identifier: doi:10.1216/JCA-2009-1-1-3
    Project Euclid: euclid.jca/1229376151
  • A. Bouvier, Anneaux présimplifiables et anneaux atomiques, C.R. Acad. Sci. Paris 272 (1971), 992–994.
    Zentralblatt MATH: 0216.32804
  • T. Dumitrescu, S.O. Ibrahim Al-Salihi, N. Radu and T. Shah, Some factorization properties of composite domains $A + XB[X]$ and $A + XB[\![X]\!]$, Comm. Alg. 28 (2000), 1125–1139.
  • S. Hizem, Chain conditions in rings of the form $A + XB[X]$ and $A + XI[X]$, pp. 259–274 in Commutative algebra and its applications, edited by M. Fontana et al., de Gruyter, Berlin (2009).
  • J. Huckaba, Commutative rings with zero divisors, Dekker, New York (1988).
    Zentralblatt MATH: 0637.13001
  • J.W. Lim and D.Y. Oh, Chain conditions in special pullbacks, C.R. Math. Acad. Sci. Paris 350 (2012), 655–659.
    Mathematical Reviews (MathSciNet): MR2971375
    Zentralblatt MATH: 1258.13025
    Digital Object Identifier: doi:10.1016/j.crma.2012.07.003
  • Z. Liu, The ascending chain condition for principal ideals of rings of generalized power series, Comm. Alg. 32 (2004), 3305–3314.
    Zentralblatt MATH: 1061.16048
    Digital Object Identifier: doi:10.1081/AGB-120039398
  • P. Ribenboim, Noetherian rings of generalized power series, J. Pure Appl. Alg. 79 (1992), 293–312.
    Zentralblatt MATH: 0761.13007
    Digital Object Identifier: doi:10.1016/0022-4049(92)90056-L
  • ––––, Rings of generalized power series II: Units and zero-divisors, J. Algebra 168 (1994), 71–89.
    Mathematical Reviews (MathSciNet): MR1289092
    Digital Object Identifier: doi:10.1006/jabr.1994.1221
  • ––––, Special properties of generalized power series, J. Algebra 173 (1995), 566–586.

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