The extension property for free modules (Q6054093)

Recall that an automorphism \(\alpha:U\rightarrow U\) in category \(C\), that is said to satisfy the \textit{extension property}, provided that, for any monomorphisms \(\lambda:U\rightarrow N\) in \(C\), there exists an automorphism \(\widetilde{\alpha}:N\rightarrow N\) such that \(\widetilde{\alpha} \lambda=\lambda \alpha\). Note that, the extension property is not true for all categories. The extension property has been studied in various categories, such as vector spaces, groups, and algebras. It is well known that, in the category of vector space, all automorphisms satisfy the extension property. \textit{P. E. Schupp} [Proc. Am. Math. Soc. 101, 226--228 (1987; Zbl 0627.20018)] proved that the automorphisms having the extension property in the category of groups, characterize the inner automorphisms. Later, \textit{L. Ben Yakoub} [Port. Math. 51, No. 2, 231--233 (1994; Zbl 0818.16028)] proved that this result is not true in the algebra category. As a generalization of Schupp's result, [\textit{S. Abdelalim} and \textit{H. Essannouni}, Port. Math. (N.S.) 59, No. 3, 325--333 (2002; Zbl 1011.20049)] characterized automorphisms having the extension property in the category of abelian groups. In this paper, let \(R\) be an integral domain not a field and let \(M\) be a free \(R\)-module, the authors give a necessary and sufficient condition such that an automorphism \(\alpha\) of \(M\) satisfies the extension property.