Linear groups over maximal orders (Q799816)

For a ring R with 1 let R-Mod be the category of right R-modules. For M in R-Mod let GL(M) be its automorphism group and let PGL(M) be the projective group of M, that is GL(M) mod. the units in the centre of R. This paper is a contribution to the study of the isomorphisms between projective groups. Suppose M and M' are right modules over the rings R and R' respectively, then any category equivalence \(F:R-Mod\to R'-Mod\) with \(F(M)=M'\) induces a group isomorphism \(F:PGL(M)\to PGL(M')\), in this case the isomorphism is said to be of equivalence type. Suppose now that M is reflexive in R-Mod and denote its dual by \(M^*\) then by transpose- inverse one obtains a group isomorphism \(PGL(M)\to PGL(M^*).\) As a consequence of the main result of this paper the author obtains: If M and M' are assumed to be finitely generated projective modules over maximal orders in central simple algebras A and A' such that both \(M\otimes A\) and M'\(\otimes A'\) have lengths \(\geq 3\), then any group isomorphism PGL(M)\(\to PGL(M')\) is either of equivalence type or the composite of a transpose-inverse isomorphism with an equivalence type isomorphism.