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Self-cancellation of torsion-free Abelian groups of finite rank. (Russian, English) Zbl 1052.20041

Zap. Nauchn. Semin. POMI 265, 7-10 (1999); translation in J. Math. Sci., New York 112, No. 3, 4247-4248 (2002).
An Abelian group \(A\) is said to have self-cancellation if \(A\oplus A\cong A\oplus B\) implies \(A\cong B\). A very simple example of a rank 4 torsion-free Abelian group without the self-cancellation property is constructed. The construction is based on the author’s criterion [Algebra Anal. 7, No. 6, 33-78 (1995); corrections ibid. 11, No. 4, 222-224 (1999; Zbl 0861.16011)] for an Abelian group \(A\) to have self-cancellation in terms of certain properties of a maximal order in the semi-simple algebra \(\mathbb{Q} R\), where \(R=\mathbb{E}(A)/N(\mathbb{E}(A))\), \(\mathbb{E}(A)\) is the endomorphism ring of \(A\) and \(N(\mathbb{E}(A))\) is the nil-radical of \(\mathbb{E}(A)\).

MSC:

20K15 Torsion-free groups, finite rank
20K25 Direct sums, direct products, etc. for abelian groups
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)

Citations:

Zbl 0861.16011
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