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

V. A. Koibaev

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 3, Tome 211 (1994), pp. 133-135

Résumé

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.