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We prove that if $G$ is an abelian $p$ -group with a nice subgroup $A$ so that $G / A$ is a $Σ$ -group, then $G$ is a $Σ$ -group if and only if $A$ is a $Σ$ -subgroup in $G$ provided that $A$ is equipped with a valuation induced by the restricted height function on $G$ . In particular, if in addition $A$ is pure in $G$ , $G$ is a $Σ$ -group precisely when $A$ is a $Σ$ -group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc. Sc. Math....

The idempotent lifting theorem for almost completely decomposable abelian groups

A. Mader, C. Vinsonhaler (1997)

Colloquium Mathematicae

The nonseparability of simply presented mixed groups

Paul Hill, Charles K. Megibben (1998)

Commentationes Mathematicae Universitatis Carolinae

It is demonstrated that an isotype subgroup of a simply presented abelian group can be simply presented without being a separable subgroup. In particular, the conjecture based on a variety of special cases that Warfield groups are absolutely separable is disproved.

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The Balance of Tor.

The $\oplus_{c}$ -topology is not completable

The elementary theory of Abelian groups with m-chains of pure subgroups

The generalized criterion of Dieudonné for primary valuated groups

The generalized criterion of Dieudonné for valuated $p$ -groups

The idempotent lifting theorem for almost completely decomposable abelian groups

The nonseparability of simply presented mixed groups

The size of an annihilator in a factorization.

The splitting of the tensor product of two mixed abelian groups of rank one

Torsion-free abelian groups and completely decomposable $p$ -adic modules

Currently displaying 1 – 10 of 10