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Summability of the ring of endomorphisms of vector groups. (English. Russian original) Zbl 0941.20063
Algebra Logika 37, No. 1, 88-100 (1998); translation in Algebra Logic 37, No. 1, 48-55 (1998).
The ring of endomorphisms \(E(G)\) of some Abelian group \(G\) is said to be summable if its automorphism group \(A(G)\) generates it additively, \(\langle A(G)\rangle=E(G)\). The question may be posed: For what kinds of Abelian groups \(G\) is the ring \(E(G)\) summable? A positive solution to this problem was given for certain classes of primary (\(p\neq 2\)) Abelian groups and for completely decomposable torsion-free Abelian groups.
Denote by \({\mathcal A}_\oplus\) the class of all Abelian groups \(G\) whose endomorphism rings are isomorphic to a ring of row-finite matrices \(M({\mathcal X})\), where \({\mathcal X}=\{\text{Hom}(G_\xi,G_\lambda)\}_{\xi,\lambda\in\Xi}\) or \({\mathcal X}=\{\text{Hom}(G_\lambda,G_\xi)\}_{\xi,\lambda\in\Xi}\) and the groups \(G_\xi\) are direct summands of \(G\) providing a simultaneous permutation of the corresponding rows and columns. The main result of the article is as follows: If, for some endomorphism \(\eta\) of the group \(G\) in \({\mathcal A}_\oplus\), the corresponding endomorphism \(\eta_{\xi,\xi}\) of the group \(G_\xi\) (\(\xi\in\Xi\)) is a sum of \(n=n(\eta)\) (\(n\geq 3\)) automorphisms, then \(\eta\) possesses this property too; consequently, \(\langle A(G)\rangle=E(G)\).
MSC:
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K25 Direct sums, direct products, etc. for abelian groups
16S50 Endomorphism rings; matrix rings
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