Commutative affine rings (Q1972171)

The semidirect extension of the automorphism group \(\Aut_RR\), 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.