We know that all automorphism \(\alpha \) of an injective module D can be extended to an automorphism of any module N which embedded D as a submodule. We say then that \(\alpha \) satisfies the extension property in the category of module. Extending an automorphism in a given category is both interesting and a difficult problem to solve. However in the category of groups, Schupp [10] proved that only inner automorphisms have the extension property. Ben Yakoub [5], in the category of algebras, has found an automorphism which is not inner but satisfies the extension property. Abdelalim et al. [1–3] solved this problem in a category of mixed modules over a bounded factorization domain, in the category of free modules over an integral domain, and in a category of torsion modules over a unique factorization domain. One of the issues that we think is important is this. What happens to the results of the extension property, if we weaken the conditions on the ring of scalars while widening the category of mixed modules? Let A be an integral Domain. Let \(M'\) be a free A-module and T a torsion A-module such that \(\bigcap \nolimits _{a \in A^{*}}aT=\{0_T\}\) . We give a necessary and sufficient condition such that an automorphism \(\alpha \) of \(M=M'\oplus T\) satisfies the extension property.
