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- W2075170368 abstract "A remarkable theorem of Butler [BTLR] says that a torsion-free abelian group is a pure subgroup of a finite direct sum of rank-one groups if and only if it is a homomorphic image of a finite direct sum of rank-one groups. There are some obvious connections between these Butler groups and the subgroups of finite direct sums of cyclic local valuated groups studied by Moore [MOOR]; for example, each rank-one torsion-free group is determined by any nonzero cyclic valuated subgroup. The juxtaposition of the two papers suggests the question: is a valuated group a subgroup of a finite direct sum of cyclics if and only if it is a homomorphic image (that is, a valuated quotient) of a finite direct sum of cyclics‘! We answer this question in the affirmative. Moore’s main result is that each subgroup of a finite direct sum of cyclic valuated groups is nice; the restriction to the local case is only apparent as subgroups are nice exactly when they are locally nice. In Section 2 we provide a simple constructive proof of a generalization of Moore’s theorem. In Section 3 we introduce, for each principal ideal domain R, the class 9$ of quotients of finite direct sums of cyclic valuated R-modules, and show that it coincides with the class of submodules of finite direct sums of cyclic valuated modules. In Section 4 we show how this result implies the equivalence of the two characterizations of Butler groups. Let R be a discrete valuation ring. Moore gave a necessary condition for a valuated R-module to be a submodule of a finite direct sum of cyclics. Section 5 contains a strengthening of this condition, and in Section 6 we characterize those discrete valuation rings R such that every rank-2 R-module that satisfies Moore’s condition is a submodule of a finite direct sum of cyclics. Stanton introduced invariants for Warlield groups which make sense for" @default.
- W2075170368 created "2016-06-24" @default.
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- W2075170368 date "1988-04-01" @default.
- W2075170368 modified "2023-09-28" @default.
- W2075170368 title "Subgroups of finite direct sums of valuated cyclic groups" @default.
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- W2075170368 doi "https://doi.org/10.1016/0021-8693(88)90207-4" @default.
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