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Modules over group rings of solvable groups with rank restrictions on subgroups

Abstract

Let A be an R G-module over a commutative ring R, where G is a group of infinite section p-rank (0-rank), C G (A) = 1, A is not a Noetherian R-module, and the quotient A/C A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (0-rank). We describe the structure of solvable groups G of this type.

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References

  1. 1.

    R. Baer and H. Heineken, “Radical groups of finite abelian subgroup rank,” Illinois J.Math. 16(4), 533–580 (1972).

    MathSciNet  MATH  Google Scholar 

  2. 2.

    O. Yu. Dashkova, “On one class of modules that are close to Noetherian,” Fundam. Prikl.Mat. 15(7), 113–125 (2009) [J.Math. Sci. 169 (5), 636–643 (2010)].

    Google Scholar 

  3. 3.

    O. Yu. Dashkova, “On modules over group rings of locally soluble groups with rank restrictions on some systems of subgroups,” Asian-Eur. J. Math. 3(1), 45–55 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    O. Yu. Dashkova, “Application of rank restrictions in the study ofmodules over integer group rings of solvable groups,” in Group Theory and its Applications (Kabardino-Balkarskiĭ Gov. Univ., Nal’chik, 2010), 77–86 [in Russian].

    Google Scholar 

  5. 5.

    O. Yu. Dashkova, M. R. Dixon, and L. A. Kurdachenko, “Linear groups with rank restrictions on the subgroups of infinite central dimension,” J. Pure Appl. Algebra 208(3), 785–795 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    M. R. Dixon, M. J. Evans, and L. A. Kurdachenko, “Linear groups with the au]minimal condition on subgroups of infinite central dimension,” J. Algebra 277(1), 172–186 (2004).

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    O. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups (North-Holland, Amsterdam-London, 1973).

    MATH  Google Scholar 

  8. 8.

    L. A. Kurdachenko, “Groups with minimax classes of conjugate elements,” in Infinite Groups and Related Algebraic Structures (Akad. Nauk Ukrainy, Inst. Mat., Kiev, 1993), 160–177 [in Russian].

    Google Scholar 

  9. 9.

    A. I. Mal’tsev, “Groups of finite rank,” Mat. Sb. 22(2), 351–352 (1948) [in Russian].

    Google Scholar 

  10. 10.

    R. E. Phillips, “The structure of groups of finitary transformations,” J. Algebra 119(2), 400–448 (1988).

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    B. A. F. Wehrfritz, Infinite Linear Groups (Springer-Verlag, New York-Heidelberg-Berlin, 1973).

    MATH  Book  Google Scholar 

  12. 12.

    B. A. F. Wehrfritz, “Artinian-finitary groups over commutative rings,” Illinois J. Math. 47(1–2), 551–565 (2003).

    MathSciNet  MATH  Google Scholar 

  13. 13.

    B. A. F. Wehrfritz, “Artinian-finitary groups over commutative rings and non-commutative rings,” J. London Math. Soc. (2) 70(2), 325–340 (2004).

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    B. A. F. Wehrfritz, “Artinian-finitary groups are locally normal-finitary,” J. Algebra 287(2), 417–431 (2005).

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to O. Yu. Dashkova.

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Original Russian Text © O. Yu. Dashkova, 2012, published in Matematicheskie Trudy, 2012, Vol. 15, No. 1, pp. 74–85.

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Dashkova, O.Y. Modules over group rings of solvable groups with rank restrictions on subgroups. Sib. Adv. Math. 23, 77–83 (2013). https://doi.org/10.3103/S1055134413020016

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Keywords

  • section p-rank
  • Noetherian module
  • solvable group