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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

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1572-9176
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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
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Published Online: 2018-01-24 | DOI: https://doi.org/10.1515/gmj-2017-0049

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

={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

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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.

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