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
R. Baer and H. Heineken, “Radical groups of finite abelian subgroup rank,” Illinois J.Math. 16(4), 533–580 (1972).
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)].
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).
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].
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).
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).
O. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups (North-Holland, Amsterdam-London, 1973).
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].
A. I. Mal’tsev, “Groups of finite rank,” Mat. Sb. 22(2), 351–352 (1948) [in Russian].
R. E. Phillips, “The structure of groups of finitary transformations,” J. Algebra 119(2), 400–448 (1988).
B. A. F. Wehrfritz, Infinite Linear Groups (Springer-Verlag, New York-Heidelberg-Berlin, 1973).
B. A. F. Wehrfritz, “Artinian-finitary groups over commutative rings,” Illinois J. Math. 47(1–2), 551–565 (2003).
B. A. F. Wehrfritz, “Artinian-finitary groups over commutative rings and non-commutative rings,” J. London Math. Soc. (2) 70(2), 325–340 (2004).
B. A. F. Wehrfritz, “Artinian-finitary groups are locally normal-finitary,” J. Algebra 287(2), 417–431 (2005).
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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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DOI: https://doi.org/10.3103/S1055134413020016