This site uses cookies. By continuing to use this site you agree to our use of cookies. To find out more, see our Privacy and Cookies policy.
Close this notification
Paper The following article is OPEN ACCESS

A Generalization of Basis and Free Modules Relatives to a Family Script U of R-Modules

, and

Published under licence by IOP Publishing Ltd
, ,



Let Script U be a family of R-modules and V be a submodule of a direct sum of some elements in Script U The aim of this paper is to generalize basis and free modules. We use the concept of Script Uv -generated module and X-sublinearly independent to provide the definition of Script U-basis and Script U-free module. We construct a Script U̲-basis of an R-module M as a pair (X, V), which a family Script U is X-sub-linearly independent to M and M is a Script Uv -generated module. Furthermore, we define Script U-basis of M as a Script U̲-basis which has the maximal element on the first component and the minimal element on the second component of a pair (X, V). The results show that the first component of (X, V) in Script U-basis is closed under submodules and intersections. Moreover, we prove that the second component of (X, V) in Script U-basis is closed under direct sums. We also determine some Script U-free modules related to a family Script U which contains all Z-module Z modulo p power of n, where p prime and n ≥ 2.

Export citation and abstract BibTeX RIS


Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.


  • [1]
    Adkins W A and Weintraub S H 1992 Algebra, An Approach via Module Theory (New York: Springer-Verlag)
  • [2]
    Dummit D S and Foote R M 2004 Abstract Algebra (USA: John Wiley and Sons, Inc.)
  • [3]
    Davvaz B and Parnian-Garamaleky Y A 1999 A Note on Exact Sequences Bull. Malaysian Math. Soc. 22 53-6
  • [4]
    Anvanriyeh S M and Davvaz B 2005 On Quasi-Exact Sequences Bull. Korean Math. Soc 42 149-55


  • [5]
    Davvaz B and Shabani-Solt H 2002 A generalization of homological algebra J. Korean Math. Soc. 39 881-98


  • [6]
    Anvanriyeh S M and Davvaz B 2002 U-Split Exact Sequences Far East J. Math. Sci. 4 209-19
  • [7]
    Madanshekaf A 2008 Quasi-Exact Sequence and Finitely Presented Modules Iran. J. Math. Sci. Informatics 3 49-53
  • [8]
    Aminizadeh R, Rasouli H and Tehranian A 2017 Quasi-exact Sequences of S-Act Bull. Malaysian Math. Soc.
  • [9]
    Fitriani, Surodjo B and Wijayanti I E 2016 On sub-exact sequences Far East J. Math. Sci. 100 1055-65
  • [10]
    Fitriani, Surodjo B and Wijayanti I E 2017 On X-sub-linearly independent modules J. Phys. Conf. Ser. 893
  • [11]
    Fitriani, Wijayanti I E and Surodjo B 2018 Generalization of U-Generator and M-Subgenerator Related to Category a [M] Journal Math. Res. 10 101-6


  • [12]
    Anderson F W and Fuller K R 1992 Rings and Categories of Modules (New York: Springer-Verlag)


  • [13]
    Wisbauer R 1991 Foundation of Module and Ring Theory (Philadelphia, USA: Gordon and Breach)
  • [14]
    Clark J, Lomp C, Vanaja N and Wisbauer R 2006 Lifting modules: supplements and projectivity in module theory (Birkhäuser Verlag)
  • [15]
    Hill V E 2000 Groups and characters (Chapman & Hall/CRC)

Export references: BibTeX RIS