Abstract.
A p-local module M is called (fully) transitive if for all \(x,y\in M\) with U M (x) = U M (y) (\(U_M(x)\leqq U_M(y)\)) there exists an automorphism (endomorphism) of M which maps x onto y. In this paper we examine the relationship of these two notions in the case of p-local modules. We show that a module M is fully transitive if and only if \(M\oplus M\) is transitive in the case where the divisible part of \(M/tM\) has rank at most one. Moreover, we show that for the same class of modules transitivity implies full transitivity if p > 2. This extends theorems of Files, Goldsmith and of Kaplansky for torsion p-local modules.
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Received: 8.2.1999
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Hennecke, G., Strüngmann, L. Transitivity and full transitivity for p-local modules. Arch. Math. 74, 321–329 (2000). https://doi.org/10.1007/s000130050450
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DOI: https://doi.org/10.1007/s000130050450