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Groups of module automorphisms and Krull dimension and codimension. (English) Zbl 1383.16023
Summary: Let \(M\) be a module over the ring \(R\) and \(G\) a subgroup of \(\mathrm{Aut}_RM\). There has been interest of late in conditions on \(R\), \(M\) and \(G\) such that one of \([M, G]\) and \(M/C_M(G)\) small ensures that the other is also small. Here we consider when the Krull dimension (resp. codimension) of one of these bounds the Krull dimension (resp. codimension) of the other. This extends earlier work where small meant Noetherian or Artinian. Unfortunately our conclusions are patchy.
MSC:
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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[1] Dixon, MR; Kurdachenko, LA; Otal, J, Linear analogues of theorems of Schur, Baer and Hall, Int. J. Group Theory, 2, 79-89, (2013) · Zbl 1306.20055
[2] Kurdachenko, LA; Subbotin, IYa; Chupordia, VA, On the relations between the central factor-module and the derived submodule in modules over group algebras, Comment. Math. Univ. Carol., 56, 433-445, (2015) · Zbl 1345.20008
[3] Wehrfritz, B.A.F.: On soluble groups of finite rank of module automorphisms, Czechoslovak Math. J. (to appear) · Zbl 06770132
[4] Wehrfritz, B.A.F.: Groups of module automorphisms of finite rank, Rend. Circ. Mat. Palermo (to appear). doi:10.1007/s12215-016-0252-z · Zbl 1380.16004
[5] McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. John Wiley and Sons, Chichester (1987) · Zbl 0644.16008
[6] Fuchs, L.: Infinite Abelian Groups, vol. 1. Academic Press, New York (1970) · Zbl 0209.05503
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