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

Yıl 2021, Cilt: 29 Sayı: 29, 107 - 119, 05.01.2021
https://doi.org/10.24330/ieja.852029

Öz

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 $R$.

Kaynakça

  • 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.
Yıl 2021, Cilt: 29 Sayı: 29, 107 - 119, 05.01.2021
https://doi.org/10.24330/ieja.852029

Öz

Kaynakça

  • 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.
Toplam 6 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Mohamed Benslımane Bu kişi benim

Hanane El Cuera Bu kişi benim

Rachid Trıbak Bu kişi benim

Yayımlanma Tarihi 5 Ocak 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 29 Sayı: 29

Kaynak Göster

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. Ocak 2021;29(29):107-119. doi:10.24330/ieja.852029
Chicago Benslımane, Mohamed, Hanane El Cuera, ve Rachid Trıbak. “MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE”. International Electronic Journal of Algebra 29, sy. 29 (Ocak 2021): 107-19. https://doi.org/10.24330/ieja.852029.
EndNote Benslımane M, El Cuera H, Trıbak R (01 Ocak 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, ve R. Trıbak, “MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE”, IEJA, c. 29, sy. 29, ss. 107–119, 2021, doi: 10.24330/ieja.852029.
ISNAD Benslımane, Mohamed vd. “MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE”. International Electronic Journal of Algebra 29/29 (Ocak 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 vd. “MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE”. International Electronic Journal of Algebra, c. 29, sy. 29, 2021, ss. 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.