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Axiom 3 modules. (English) Zbl 0597.20048

This paper introduces an important new tool for use in the study of abelian groups - the concept of a knice submodule. Let G be a p-local abelian group and \({\bar \alpha}=\{\alpha_ n|\) \(n<\omega \}^ a \)height sequence. Denote
G(\({\bar \alpha}\))\(=\{x\in G|\) \(height(p^ nx)\geq \alpha_ n\) for all \(n<\omega \}\)
G(\({\bar \alpha}{}^*)=<\{x\in G({\bar \alpha})|\) \(height(p^ nx)>\alpha_ n\) for infinitely many \(n\}>.\)
By convention, G(\({\bar \alpha}{}^*)\) is the torsion submodule of G(\({\bar \alpha}\)) in case \(\alpha_ n=\infty\) for some n. An element x of G is called primitive if x has height sequence \({\bar \alpha}\) and \(x\not\in G({\bar \alpha}^*)\). Finally, a direct sum \(A_ 1\oplus...\oplus A_ n\) of submodules of G is called a *-valuated coproduct if for \(a_ i\in A_ i\), \(height(a_ 1+...+a_ n)=\inf (height(a_ i)\}\), and for all height sequences \({\bar \alpha}\), \(a_ 1+...+a_ n\in G({\bar \alpha}^*)\) implies each \(a_ i\in G({\bar \alpha}^*).\)
A submodule N of G is said to be knice provided the following two conditions are satisfied.
(1) N is nice in G
(2) If S is a finite subset of G, then there is a finite collection of primitive elements \(y_ 1,...,y_ m\) and an \(r<\omega\) such that \(N\oplus <y_ 1>\oplus...\oplus <y_ n>\) is a *-valuated coproduct that contains \(p^ r<S>\). In case G is torsion, knice reduces to nice.
The p-local group G is called an Axiom 3 module if G satisfies the third axiom of countability with respect to knice submodules. The authors show that Axioms 3 modules are characterized by their Ulm-Kaplansky invariants and their Warfield invariants. They then show that the Axiom 3 modules are precisely the local Warfield modules - direct summands of simply presented \(Z_ p\)-modules. The Axiom 3 characterization is used to solve a ten year old problem of Warfield:
Theorem 4.2. A p-local group G is an Axiom 3 module if and only if G has a decomposition basis and satisfies the third axiom of countability with respect to nice submodules.
The final section contains two theorems giving sufficient conditions for G to be an Axioms 3 module, using an extension of the definition of isotype, and the Kulikov-Megibben criterion, respectively.
Reviewer: C.Vinsonhaler

MSC:

20K21 Mixed groups
20K10 Torsion groups, primary groups and generalized primary groups
20K27 Subgroups of abelian groups
20K99 Abelian groups
20K25 Direct sums, direct products, etc. for abelian groups
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