Corings and invertible bimodules (Q1812862)

Let \(S\subset R\) be (non-commutative) rings. \(\text{Inv}_ S(R)\) is the group of invertible \(S\)-subbimodules of \(R\) and \(\Aut_{R\text{- cor}}(R\otimes_ S R)\) is the group of \(R\)-coring automorphisms of \(R \otimes_ S R\). The author defines a natural group map \(\Gamma: \text{Inv}_ S(R)\to\Aut_{R\text{-cor}}(R \otimes_ S R)\) and proves that \(\Gamma\) is an isomorphism, if either \(R\) is faithfully flat as a right or left \(S\)-module or \(S\) is a direct summand as a right \(S\)-module and the functor \(-\otimes_ SR\) reflects isomorphisms.