Home On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules
Article
Licensed
Unlicensed Requires Authentication

On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules

  • Shahram Motmaen and Ahmad Yousefian Darani EMAIL logo
Published/Copyright: January 24, 2018
Become an author with De Gruyter Brill
Author Information Explore this Subject
Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 27 Issue 1

Abstract

In this paper, we introduce some classes of R-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set

ℍ={M∣M⁢ is an ⁢R⁢-module and ⁢Nil⁢(M)⁢ is a divided prime submodule of ⁢M}.

For an R-module M∈ℍ, set T=(R∖Z⁢(R))∩(R∖Z⁢(M)), 𝔗⁢(M)=T-1⁢M and P=(Nil(M):RM). In this case, the mapping Φ:𝔗⁢(M)→MP given by Φ⁢(x/s)=x/s is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M into MP given by Φ⁢(x)=x/1 for every x∈M. A nonnil submodule N of M is said to be Φ-invertible if Φ⁢(N) is an invertible submodule of Φ⁢(M). Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if Φ⁢(N) is a principal ideal of Φ⁢(M) for every finitely generated submodule N of the R-module M. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R-modules.

Keywords: Dedekind ring; Dedekind module; Prüfer ring; Prüfer module
MSC 2010: 13A05; 13F05

References

[1] M. M. Ali, Invertibility of multiplication modules, New Zealand J. Math. 35 (2006), no. 1, 17–29. Search in Google Scholar

[2] M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, Comm. Algebra 36 (2008), no. 12, 4620–4642. 10.1080/00927870802186805Search in Google Scholar

[3] M. M. Ali, Invertibility of multiplication modules II, New Zealand J. Math. 39 (2009), 45–64. Search in Google Scholar

[4] M. M. Ali, Invertibility of multiplication modules III, New Zealand J. Math. 39 (2009), 193–213. Search in Google Scholar

[5] D. D. Anderson, Some remarks on multiplication ideals. II, Comm. Algebra 28 (2000), no. 5, 2577–2583. 10.1080/00927870008826980Search in Google Scholar

[6] D. F. Anderson and A. Badawi, On ϕ-Prüfer rings and ϕ-Bezout rings, Houston J. Math. 30 (2004), no. 2, 331–343. Search in Google Scholar

[7] D. F. Anderson and A. Badawi, On ϕ-Dedekind rings and ϕ-Krull rings, Houston J. Math. 31 (2005), no. 4, 1007–1022. Search in Google Scholar

[8] A. Badawi, On divided commutative rings, Comm. Algebra 27 (1999), no. 3, 1465–1474. 10.1080/00927879908826507Search in Google Scholar

[9] A. Badawi, On ϕ-pseudo-valuation rings, Advances in Commutative Ring Theory (Fez 1997), Lect. Notes Pure Appl. Math. 205, Dekker, New York (1999), 101–110. 10.1201/9781003419815-7Search in Google Scholar

[10] A. Badawi, On Φ-pseudo-valuation rings. II, Houston J. Math. 26 (2000), no. 3, 473–480. Search in Google Scholar

[11] A. Badawi, On ϕ-chained rings and ϕ-pseudo-valuation rings, Houston J. Math. 27 (2001), no. 4, 725–736. Search in Google Scholar

[12] A. Badawi, On divided rings and ϕ-pseudo-valuation rings, Internat. J. Commut. Rings 1 (2002), no. 2, 51–60. Search in Google Scholar

[13] A. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31 (2003), no. 4, 1669–1677. 10.1081/AGB-120018502Search in Google Scholar

[14] A. Badawi and T. G. Lucas, Rings with prime nilradical, Arithmetical Properties of Commutative Rings and Monoids, Lect. Notes Pure Appl. Math. 241, Chapman & Hall/CRC, Boca Raton, FL (2005), 198–212. 10.1201/9781420028249.ch11Search in Google Scholar

[15] A. Badawi and T. G. Lucas, On Φ-Mori rings, Houston J. Math. 32 (2006), no. 1, 1–32. Search in Google Scholar

[16] D. E. Dobbs, Divided rings and going-down, Pacific J. Math. 67 (1976), no. 2, 353–363. 10.2140/pjm.1976.67.353Search in Google Scholar

[17] Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra 16 (1988), no. 4, 755–779. 10.1080/00927878808823601Search in Google Scholar

[18] A. G. Naoum and F. H. Al-Alwan, Dedekind modules, Comm. Algebra 24 (1996), no. 2, 397–412. 10.1080/00927879608825576Search in Google Scholar

[19] R. Y. Sharp, Steps in Commutative Algebra, London Math. Soc. Stud. Texts 19, Cambridge University Press, Cambridge, 1990. Search in Google Scholar

Received: 2015-04-18
Revised: 2016-02-25
Accepted: 2016-05-06
Published Online: 2018-01-24
Published in Print: 2020-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials
  3. On generalized α-ψ-Geraghty contractions on b-metric spaces
  4. New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods
  5. A Tauberian theorem for the generalized Nörlund summability method
  6. A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities
  7. The Robin function and conformal welding – A new proof of the existence
  8. Effects of the initial moment and several delays perturbations in the variation formulas for a solution of a functional differential equation with the continuous initial condition
  9. The well-posedness of a nonlocal multipoint problem for a differential operator equation of second order
  10. Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations
  11. On an approximate solution of a class of surface singular integral equations of the first kind
  12. On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules
  13. On the geometrical properties of hypercomplex four-dimensional Lie groups
  14. On sets of singular rotations for translation invariant differentiation bases formed by intervals
  15. Certain commutativity criteria for rings with involution involving generalized derivations
  16. The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds
  17. Ripplet transform and its extension to Boehmians
  18. Variable exponent fractional integrals in the limiting case α(x)p(n) ≡ n on quasimetric measure spaces
https://doi.org/10.1515/gmj-2017-0049
Keywords for this article
Dedekind ring; Dedekind module; Prüfer ring; Prüfer module

Articles in the same Issue

  1. Frontmatter
  2. An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials
  3. On generalized α-ψ-Geraghty contractions on b-metric spaces
  4. New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods
  5. A Tauberian theorem for the generalized Nörlund summability method
  6. A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities
  7. The Robin function and conformal welding – A new proof of the existence
  8. Effects of the initial moment and several delays perturbations in the variation formulas for a solution of a functional differential equation with the continuous initial condition
  9. The well-posedness of a nonlocal multipoint problem for a differential operator equation of second order
  10. Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations
  11. On an approximate solution of a class of surface singular integral equations of the first kind
  12. On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules
  13. On the geometrical properties of hypercomplex four-dimensional Lie groups
  14. On sets of singular rotations for translation invariant differentiation bases formed by intervals
  15. Certain commutativity criteria for rings with involution involving generalized derivations
  16. The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds
  17. Ripplet transform and its extension to Boehmians
  18. Variable exponent fractional integrals in the limiting case α(x)p(n) ≡ n on quasimetric measure spaces
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 9.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2017-0049/html
Scroll to top button