The extension property for free modules

Describing automorphisms having the extension property in any category is a difficult problem. Nevertheless there are very important results. In the category of groups, Schupp (Proc Am Math Soc 101(2):226-228, 1987) proved that only inner automorphisms have the extension property. In order to generalize Schupp's result, Abdelalim and Essannouni (Port Math 59(3):325-334, 2002) characterized the automorphisms having the extension property, in the category of abelians groups. It was therefore legitimate to work in categories containing that of abelian groups. One of the issues that we think is important is the following: what can be done about the extension property in the category of free modules over just an integral domain? Let A be an integral domain not a field and let M be a free A-module, we give a necessary and sufficient condition such that an automorphism alpha of M satisfies the extension property.