The normalizer of the automorphism group of a module arising under extension of the base ring

Let $\Lambda$ te an arbitrary associative ring with unity and let $R$ be its unital subring contained in the center of $\Lambda$. Further, let $M=_\Lambda M$ be a left free $\Lambda$-module of finite rank. In this paper, the normalizer of the subgroup $\mathrm{Aut}(_\Lambda M)$ of automorphisms of the module $_\Lambda M$ in the group $\mathrm{Aut}(_RM)$ of automorphisms of the moduleRM is computed. If the ring $\Lambda$ is additively generated by its invertible elements, then the above normalizer coincides with the semidirect product of the normal subgroup $\mathrm{Aut}(_\Lambda M)$ and a subgroup isomorphic to the group $\mathrm{Aut}(\Lambda/R)$ of all ring automorphisms of the ring $\Lambda$ that are identical on $R$. Bibliography: 1 title.