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MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE

Year 2021, , 107 - 119, 05.01.2021
https://doi.org/10.24330/ieja.852029

Abstract

All rings are commutative. Let $M$ be a module. We introduce the property $({\bf P})$: Every endomorphism of $M$ has a non-trivial invariant submodule. We determine the structure of all vector spaces having $({\bf P})$ over any field and all semisimple modules satisfying $({\bf P})$ over any ring. Also, we provide a structure theorem for abelian groups having this property. We conclude the paper by characterizing the class of rings for which every module satisfies $({\bf P})$ as that of the rings $R$ for which $R/\mathfrak{m}$ is an algebraically closed field for every maximal ideal $\mathfrak{m}$ of $R$.

References

  • N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math., 60(2) (1954), 345-350.
  • A. R. Bernstein and A. Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific J. Math., 16(3) (1966), 421-431.
  • R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Revised edition, The Press Syndicate of the University of Cambridge, Cambridge University Press, Cambridge, 1994.
  • M. Liu, The invariant subspace problem and its main developments, Int. J. Open Problems Compt. Math., 3(5) (2010), 88-97.
  • V. I. Lomonosov, Invariant subspaces for the family of operators which commute with a completely continuous operator, Funct. Anal. Appl., 7(3) (1973), 213-214.
  • A. C . Ozcan, A. Harmanci and P. F. Smith, Duo modules, Glasg. Math. J., 48(3) (2006), 533-545.
Year 2021, , 107 - 119, 05.01.2021
https://doi.org/10.24330/ieja.852029

Abstract

References

  • N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math., 60(2) (1954), 345-350.
  • A. R. Bernstein and A. Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific J. Math., 16(3) (1966), 421-431.
  • R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Revised edition, The Press Syndicate of the University of Cambridge, Cambridge University Press, Cambridge, 1994.
  • M. Liu, The invariant subspace problem and its main developments, Int. J. Open Problems Compt. Math., 3(5) (2010), 88-97.
  • V. I. Lomonosov, Invariant subspaces for the family of operators which commute with a completely continuous operator, Funct. Anal. Appl., 7(3) (1973), 213-214.
  • A. C . Ozcan, A. Harmanci and P. F. Smith, Duo modules, Glasg. Math. J., 48(3) (2006), 533-545.
There are 6 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mohamed Benslımane This is me

Hanane El Cuera This is me

Rachid Trıbak This is me

Publication Date January 5, 2021
Published in Issue Year 2021

Cite

APA Benslımane, M., El Cuera, H., & Trıbak, R. (2021). MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE. International Electronic Journal of Algebra, 29(29), 107-119. https://doi.org/10.24330/ieja.852029
AMA Benslımane M, El Cuera H, Trıbak R. MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE. IEJA. January 2021;29(29):107-119. doi:10.24330/ieja.852029
Chicago Benslımane, Mohamed, Hanane El Cuera, and Rachid Trıbak. “MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE”. International Electronic Journal of Algebra 29, no. 29 (January 2021): 107-19. https://doi.org/10.24330/ieja.852029.
EndNote Benslımane M, El Cuera H, Trıbak R (January 1, 2021) MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE. International Electronic Journal of Algebra 29 29 107–119.
IEEE M. Benslımane, H. El Cuera, and R. Trıbak, “MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE”, IEJA, vol. 29, no. 29, pp. 107–119, 2021, doi: 10.24330/ieja.852029.
ISNAD Benslımane, Mohamed et al. “MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE”. International Electronic Journal of Algebra 29/29 (January 2021), 107-119. https://doi.org/10.24330/ieja.852029.
JAMA Benslımane M, El Cuera H, Trıbak R. MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE. IEJA. 2021;29:107–119.
MLA Benslımane, Mohamed et al. “MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE”. International Electronic Journal of Algebra, vol. 29, no. 29, 2021, pp. 107-19, doi:10.24330/ieja.852029.
Vancouver Benslımane M, El Cuera H, Trıbak R. MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE. IEJA. 2021;29(29):107-19.