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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.

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
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References:

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