Bekker, I. Kh. Commutative affine rings. (English. Russian original) Zbl 0941.16003 Sib. Math. J. 40, No. 1, 3-8 (1999); translation from Sib. Mat. Zh. 40, No. 1, 6-13 (1999). The semidirect extension of the automorphism group \(\operatorname{Aut}_RR\), where \(R\) is an associative ring with unity, by the additive group \(R^+\) is called the affine group of the ring \(R\) and denoted by \(\text{Aff}_RR\). The author discusses the question of determining a ring \(R\) from its affine group \(\text{Aff}_RR\) in the category of \(R\)-modules. He calls a module \(_RM\) over a commutative ring \(R\) an \(E\)-module if \(\text{Hom}_\mathbb{Z}(R,M)=\text{Hom}_R(R,M)\). A ring \(R\) is called an \(E\)-ring if \(_RR\) is an \(E\)-module. The author obtains some conditions sufficient for an \(E\)-ring to be an affine ring. Reviewer: K.N.Ponomarev (Novosibirsk) Cited in 2 Documents MSC: 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16D90 Module categories in associative algebras 16S50 Endomorphism rings; matrix rings 16W20 Automorphisms and endomorphisms 20K15 Torsion-free groups, finite rank 20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups 13B10 Morphisms of commutative rings 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations Keywords:affine groups of rings; affine \(E\)-rings; automorphism groups of rings; additive groups of rings; categories of modules; \(E\)-modules PDFBibTeX XMLCite \textit{I. Kh. Bekker}, Sib. Math. J. 40, No. 1, 6--13 (1999; Zbl 0941.16003); translation from Sib. Mat. Zh. 40, No. 1, 6--13 (1999) Full Text: DOI References: [1] H. Bass, AlgebraicK-Theory [Russian translation], Mir, Moscow (1973). [2] A. I. Kostrikin and Yu. I. Manin, Linear Algebra and Geometry [in Russian], Nauka, Moscow (1986). [3] L. Fuchs, Infinite Abelian Groups. Vol. 1 [Russian translation], Mir, Moscow (1974). · Zbl 0274.20067 [4] L. Fuchs, Abelian Groups, Acad. Kiado, Budapest (1966). · Zbl 0100.02803 [5] I. Kh. Bekker and S. F. Kozhukhov, Automorphisms of Torsion-Free Abelian Groups [in Russian], Tomsk. Univ., Tomsk (1988). · Zbl 0681.20037 [6] R. S. Pierce,E-Modules, Contemp. Math.,87, 221–240 (1989). [7] M. Dugas, A. Mader, and C. Vinsonhaler, ”LargeE-rings exist,” J. Algebra,108, 88–101 (1987). · Zbl 0616.20026 [8] C. Faith, Algebra: Rings, Modules, and Categories. Vol. 1 [Russian translation], Mir, Moscow (1977). · Zbl 0345.16005 [9] A. Mader and C. Vinsonhaler, ”Torsion-freeE-modules,” J. Algebra,115, No. 2, 401–411 (1988). · Zbl 0639.13010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.