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Algebra Logika, 2009, Volume 48, Number 4, Pages 520–539 (Mi al411)  

This article is cited in 22 scientific papers (total in 22 papers)

Abelian groups with normal endomorphism rings

A. R. Chekhlov


Abstract: A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group $A$ with a normal endomorphism ring contains a pure fully invariant subgroup $G\oplus B$, the endomorphism ring of a group $G$ is commutative, and a subgroup $B$ is not always distinguished by a direct summand in $A$. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.

Keywords: fully invariant subgroup, central invariant subgroup, normal endomorphism ring, invariant endomorphism ring, Lie bracket of endomorphisms.

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English version:
Algebra and Logic, 2009, 48:4, 298–308

Bibliographic databases:

UDC: 512.541
Received: 19.01.2009
Revised: 19.02.2009

Citation: A. R. Chekhlov, “Abelian groups with normal endomorphism rings”, Algebra Logika, 48:4 (2009), 520–539; Algebra and Logic, 48:4 (2009), 298–308

Citation in format AMSBIB
\by A.~R.~Chekhlov
\paper Abelian groups with normal endomorphism rings
\jour Algebra Logika
\yr 2009
\vol 48
\issue 4
\pages 520--539
\jour Algebra and Logic
\yr 2009
\vol 48
\issue 4
\pages 298--308

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    This publication is cited in the following articles:
    1. A. R. Chekhlov, “E-nilpotentnye i E-razreshimye abelevy gruppy klassa 2”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2010, no. 1(9), 59–71  mathnet
    2. A. R. Chekhlov, “Nekotorye primery E-razreshimykh grupp”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2010, no. 3(11), 69–76  mathnet
    3. A. R. Chekhlov, “Commutator invariant subgroups of abelian groups”, Siberian Math. J., 51:5 (2010), 926–934  mathnet  crossref  mathscinet  isi
    4. A. R. Chekhlov, “$E$-solvable modules”, J. Math. Sci., 183:3 (2012), 424–434  mathnet  crossref  mathscinet
    5. A. R. Chekhlov, “O skobke Li endomorfizmov abelevykh grupp, 2”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2011, no. 1(13), 55–60  mathnet
    6. A. R. Chekhlov, “On Some Classes of Nilgroups”, Math. Notes, 91:2 (2012), 283–289  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. A. R. Chekhlov, “On the projective commutant of abelian groups”, Siberian Math. J., 53:2 (2012), 361–370  mathnet  crossref  mathscinet  isi
    8. A. R. Chekhlov, “E-engelevy abelevy gruppy stupeni $\le2$”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2012, no. 1(17), 54–60  mathnet
    9. A. R. Chekhlov, “Abelian groups with nilpotent commutators of endomorphisms”, Russian Math. (Iz. VUZ), 56:10 (2012), 50–61  mathnet  crossref  mathscinet
    10. A. R. Chekhlov, “On projectively soluble abelian groups”, Siberian Math. J., 53:5 (2012), 927–933  mathnet  crossref  mathscinet  isi
    11. A. R. Chekhlov, “On Abelian groups close to $E$-solvable groups”, J. Math. Sci., 197:5 (2014), 708–733  mathnet  crossref
    12. A. R. Chekhlov, “Torsion-Free Weakly Transitive $E$-Engel Abelian Groups”, Math. Notes, 94:4 (2013), 583–589  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. A. R. Chekhlov, “O pryamykh summakh tsiklicheskikh grupp s invariantnymi monomorfizmami”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2013, no. 3(23), 60–65  mathnet  elib
    14. A. R. Chekhlov, Ml. V. Agafontseva, “Ob abelevykh gruppakh s tsentralnymi kvadratami kommutatorov endomorfizmov”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2013, no. 4(24), 54–59  mathnet  elib
    15. A. R. Chekhlov, “On abelian groups with commuting monomorphisms”, Siberian Math. J., 54:5 (2013), 946–950  mathnet  crossref  mathscinet  isi
    16. A. R. Chekhlov, “On abelian groups with right-invariant isometries”, Siberian Math. J., 55:3 (2014), 574–577  mathnet  crossref  mathscinet  isi  elib  elib
    17. K. S. Sorokin, “$SP$-gruppy s chistymi koltsami endomorfizmov”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2014, no. 4(30), 24–35  mathnet
    18. A. R. Chekhlov, “On Abelian groups with commutative commutators of endomorphisms”, J. Math. Sci., 230:3 (2018), 502–506  mathnet  crossref  mathscinet  elib
    19. Fuchs L., “Abelian Groups”, Abelian Groups, Springer Monographs in Mathematics, Springer, 2015, 1–747  crossref  mathscinet  isi
    20. V. M. Misyakov, “Abelevy gruppy s regulyarnym tsentrom koltsa endomorfizmov”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2016, no. 2(40), 33–36  mathnet  crossref  elib
    21. Chekhlov A., Danchev P., “A Generalization of Co-Hopfian Abelian Groups”, Int. J. Algebr. Comput., 27:4 (2017), 351–360  crossref  mathscinet  zmath  isi  scopus
    22. A. R. Chekhlov, “Abelian groups with monomorphisms invariant with respect to epimorphisms”, Russian Math. (Iz. VUZ), 62:12 (2018), 74–80  mathnet  crossref  isi
  • Алгебра и логика Algebra and Logic
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