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 $R$.
Anahtar Kelimeler
Algebraically closed field characteristic polynomial of a matrix (an endomorphism) fully invariant submodule homogeneous semisimple module invariant submodule
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.
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.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 5 Ocak 2021 |
| Yayımlandığı Sayı | Yıl 2021 |