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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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