On endomorphism rings of non-separable Abelian \(p\)-groups. (English) Zbl 0687.20050
A homomorphism \(\Theta:G\to H\) between Abelian \(p\)-groups \(G, H\) is thin (a notion due to Corner 1976) if for each natural number \(e\) there exists an \(n\) such that \((p^ nG[p^ e])\Theta \subseteq p^{\omega}H\). The author constructs Abelian \(p\)-groups \(G\) of \(p\)-length \(\omega +1\) such that \(\text{End\,}G=A\oplus E_{\Theta}G\), where the ring \(A\) and the cardinality \(\lambda =\lambda^{\aleph_ 0}\geq | A|\) are prescribed. In this setting \(E_{\Theta}G\) denotes all thin endomorphisms of \(G\) and \(\oplus\) is a ring split extension.
The proof is based on a lifting technique using a similar result for Abelian separable \(p\)-groups. The author also discusses consequences under \(V=L\), however this proof seems not to work for length larger than or equal to \(\omega +\omega\) (as claimed without proof).
The proof is based on a lifting technique using a similar result for Abelian separable \(p\)-groups. The author also discusses consequences under \(V=L\), however this proof seems not to work for length larger than or equal to \(\omega +\omega\) (as claimed without proof).
Reviewer: R.Göbel
MSC:
| 20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
| 20K10 | Torsion groups, primary groups and generalized primary groups |
| 16S50 | Endomorphism rings; matrix rings |
Keywords:
realization problem; Abelian \(p\)-groups; \(p\)-length; thin endomorphisms; separable \(p\)-groups; \(V=L\)References:
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