Stacked bases for homogeneous completely decomposable groups. (English) Zbl 1019.20022

Salce, L.; Strüngmann, L.

Commun. Algebra 29, No. 6, 2575-2588 (2001).

Let \(R\) be a subgroup of the rationals containing the ring of integers and \(R_0\) the maximal subring of \(R\). An \(R\)-representation of an (Abelian) group \(G\) is an exact sequence \(0\to A\to X\to G\to 0\) such that \(A\) and \(X\) are homogeneous completely decomposable groups of type \(R\). A result of P. Hill and C. Megibben [Trans. Am. Math. Soc. 312, No. 1, 377-402 (1989; Zbl 0668.20043)] on equivalence of \(R\)-representations of \(R_0\)-modules is generalized from the case \(R=R_0\) to arbitrary \(R\). Another result states that there exists a stacked basis for \(A\subset X\) (i.e. a decomposition \(X=\bigoplus x_iR\) such that \(A=\bigoplus r_ix_iR\) for some \(r_i\in R_0)\) iff \(X/A\) is a direct sum of cyclic torsion \(R_0\)-modules and a homogeneous completely decomposable group of type \(R\).

Reviewer: Stanisław Balcerzyk (Toruń)

MSC:

20K25	Direct sums, direct products, etc. for abelian groups
20K15	Torsion-free groups, finite rank

Keywords:

subgroups of the rationals; maximal subrings; homogeneous completely decomposable groups; \(R\)-representations; stacked bases; direct sums

Citations:

Zbl 0668.20043

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