Taiwanese Journal of Mathematics

$S$-Noetherian Rings and Their Extensions

Jongwook Baeck, Gangyong Lee, and Jung Wook Lim

Full-text: Open access

Abstract

Let $R$ be an associative ring with identity, $S$ a multiplicative subset of $R$, and $M$ a right $R$-module. Then $M$ is called an $S$-Noetherian module if for each submodule $N$ of $M$, there exist an element $s \in S$ and a finitely generated submodule $F$ of $M$ such that $Ns \subseteq F \subseteq N$, and $R$ is called a right $S$-Noetherian ring if $R_R$ is an $S$-Noetherian module. In this paper, we study some properties of right $S$-Noetherian rings and $S$-Noetherian modules. Among other things, we study Ore extensions, skew-Laurent polynomial ring extensions, and power series ring extensions of $S$-Noetherian rings.

Article information

Source
Taiwanese J. Math., Volume 20, Number 6 (2016), 1231-1250.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874529

Digital Object Identifier
doi:10.11650/tjm.20.2016.7436

Mathematical Reviews number (MathSciNet)
MR3580293

Zentralblatt MATH identifier
1357.16039

Subjects
Primary: 16D25: Ideals 16P99: None of the above, but in this section 16S36: Ordinary and skew polynomial rings and semigroup rings [See also 20M25] 16U20: Ore rings, multiplicative sets, Ore localization

Keywords
right $S$-Noetherian ring $S$-Noetherian module Hilbert basis theorem Ore extension

Citation

Baeck, Jongwook; Lee, Gangyong; Lim, Jung Wook. $S$-Noetherian Rings and Their Extensions. Taiwanese J. Math. 20 (2016), no. 6, 1231--1250. doi:10.11650/tjm.20.2016.7436. https://projecteuclid.org/euclid.twjm/1498874529


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References

  • D. D. Anderson and T. Dumitrescu, $S$-Noetherian rings, Comm. Algebra 30 (2002), no. 9, 4407–4416.
  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics 13, Springer-Verlag, New York, 1974.
  • D. D. Anderson, D. J. Kwak and M. Zafrullah, Agreeable domains, Comm. Algebra 23 (1995), no. 13, 4861–4883.
  • K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Second edition, London Mathematical Society Student Texts 61, Cambridge University Press, Cambridge, 2004.
  • T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, New York, 1999.
  • ––––, A First Course in Noncommutative Rings, Second edition, Graduate Texts in Mathematics 131, Springer-Verlag, New York, 2001.
  • T.-K. Lee and Y. Zhou, Armendariz and reduced rings, Comm. Algebra 32 (2004), no. 6, 2287–2299.
  • J. W. Lim, A note on $S$-Noetherian domains, Kyungpook Math. J. 55 (2015), no. 3, 507–514.
  • J. W. Lim and D. Y. Oh, $S$-Noetherian properties on amalgamated algebras along an ideal, J. Pure Appl. Algebra 218 (2014), no. 6, 1075–1080.
  • ––––, $S$-Noetherian properties of composite ring extensions, Comm. Algebra 43 (2015), no. 7, 2820–2829.
  • Z. Liu, On $S$-Noetherian rings, Arch. Math. (Brno) 43 (2007), no. 1, 55–60.