P. V. Danchev : The Generalized Criterion of Dieudonné for Valuated Abelian Groups, p.149-155

Abstract:

An extension in terms of $\sigma_{\lambda}$-summable (valuated) groups for a cofinal with $\omega$ ordinal $\lambda$ of the classical Dieudonné criterion (Portugaliae Mathematicae, 1952), concerning the direct sums of $p$-primary cyclic groups, is established. Specifically, it is proved that if $G$ is an abelian $p$-group whose length $\lambda$ is cofinal with $\omega$ and if $A$ is a subgroup of $G$ so that $A$ is a $\sigma_{\lambda}$-summable valuated group using the valuation inherited from the height valuation on $G$ and so that $G/A$ is a $\sigma_{\lambda}$-summable group, then $G$ is a $\sigma_{\lambda}$-summable group.

Key Words: $\sigma_{\lambda}$-summable groups, heights, valuated subgroups.

2000 Mathematics Subject Classification: Primary: 20K10,
Secondary: 20K07.

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