Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2017: 0.23

See all formats and pricing
More options …
Ahead of print


On Φ-Dedekind, Φ-Pr\"ufer and Φ-Bezout modules

Shahram Motmaen / Ahmad Yousefian Darani
  • Corresponding author
  • Department of Mathematics and Applications, University of Mohaghegh Ardabili, P. O. Box 179, Ardabil, Iran
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-01-24 | DOI: https://doi.org/10.1515/gmj-2017-0049


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

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

For an R-module M, set T=(RZ(R))(RZ(M)), 𝔗(M)=T-1M 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 xM. 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


  • [1]

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

  • [2]

    M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, Comm. Algebra 36 (2008), no. 12, 4620–4642. CrossrefGoogle Scholar

  • [3]

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

  • [4]

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

  • [5]

    D. D. Anderson, Some remarks on multiplication ideals. II, Comm. Algebra 28 (2000), no. 5, 2577–2583. CrossrefGoogle 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. 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. Google Scholar

  • [8]

    A. Badawi, On divided commutative rings, Comm. Algebra 27 (1999), no. 3, 1465–1474. CrossrefGoogle 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. Google Scholar

  • [10]

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

  • [11]

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

  • [12]

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

  • [13]

    A. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31 (2003), no. 4, 1669–1677. CrossrefGoogle 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. Google Scholar

  • [15]

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

  • [16]

    D. E. Dobbs, Divided rings and going-down, Pacific J. Math. 67 (1976), no. 2, 353–363. CrossrefGoogle Scholar

  • [17]

    Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra 16 (1988), no. 4, 755–779. CrossrefGoogle Scholar

  • [18]

    A. G. Naoum and F. H. Al-Alwan, Dedekind modules, Comm. Algebra 24 (1996), no. 2, 397–412. CrossrefGoogle Scholar

  • [19]

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

About the article

Received: 2015-04-18

Revised: 2016-02-25

Accepted: 2016-05-06

Published Online: 2018-01-24

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2017-0049.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in