The Normalizer of the Elementary Linear Group of a Module Arising when the Base Ring is Extended

Abstract

Let S be a commutative ring with 1 and R a unital subring. Let M be a free S-module of rank n ≥ 3. In 1994, V. A. Koibaev described the normalizer of Aut_S(M) in the group Aut_R(M). In the present paper, it is proved that the normalizer of the elementary linear group E_????(M) in Aut_R(M) coincides with that of Aut_S(M), namely, N_AutR(M)(E_????(M)) = Aut(S/R)⋉Aut_S(M). If S is free of rank m as an R-module, then N_GL(mn,R)(E(n, S)) = Aut(S/R)⋉GL(n, S). Moreover, for any proper ideal A of R,

\( {N}_{GL\left( mn,R\right)}\left(E\left(n,S\right)E\left( mn,R,A\right)\right)={\rho}_A^{-1}\left({N}_{GL\left( mn,R/A\right)}\left(E\left(n,S/ SA\right)\right)\right). \)