THE NORMALIZER OF THE ELEMENTARY LINEAR GROUP OF A MODULE ARISING WHEN THE BASE RING IS EXTENDED

摘要

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 ~(Aut R ( 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 mn R E n S E mn R A = ρ A ? 1 N GL mn R / A E n S / SA . $$ {N}_{GLleft( mn,Rright)}left(Eleft(n,Sright)Eleft( mn,R,Aright)right)={rho}_A^{-1}left({N}_{GLleft( mn,R/Aright)}left(Eleft(n,S/ SAright)right)right). $$.