About linear groups over rings

Abstract

1. The (~-Radical. We shall prove in this section the following theorem. THEOREM I. An exterior radical of the group r is a Z-group. LEMMA 1. ff K is a commutative ring and R is its radical, then there exists in R a system of ideals of the ring K, annihilating in R. We ay that a system {~B} of ideals of a ring K is annihila*ing in R if for any jump RBc Rfl+! the inclusions R/3+vR cR~, R-R~+~cRfl hold. m ca~,e of a commutative ring K one il)clusion is, obviously, sufficient. Let {R~ be a composition Sy.~tem of all- right ideals of the ring K contained in R. Since K is a commutative ring, Rfl are even twosided idea~s in K. We shall examine the composition p~" Rfl, Rfl+ 1, R~ c Rfl+ I. The factor ring Rfl+I/Rf~ canbe considered as a right K-module, moreover, since the pai. Rfl, R- - RB c Rfl+. is a composition pair, it is an irreducible module. Jacobson radical consists by definition .~+1, 1 [2] of such and only sdch elements of the ring K, which act as zeroa in every irreducible K-module. Therefore, the inclusion R-Rfl+ic Rfl holds, hence the. system {Bfl} is annihilating. Proof of Theorem 1. We shall denote by H the subalgebra of Kn generated by the set (~(F) -F. We shaIl show that the algebra H has an annihilating system. If J is a primitive ideal of the ring K, and JK n is the corresponding submodule (jgn consists of all rows with elements from the ideal J}, then the factor-module Kn/JK n is at. n-dimensional vector space over the field K/J. Let Hj be the kernel of the rePresentation (Kn/~rK n, H}. The algebra H acts by definition as a zero in every F-composition f~ or. Since the length of a r-composition series in Kn/JK n does not exceed n, it follows that the algebra H/Hj is a nilpotcnt algebra of rank -< n-1. Let {J~}a~A be the system of all primitive ideals of the ring K. Then N J(~ = R hence 0 Hj -- H N Rn. o~EA cr Remak's theorem implies now that the algebra H/H N R n is nilpotent, m will denote the degree of nilpotency of the algebra H/H N R n. The subalgebra H N R n has an annihilating system in virtue ef Lemma 1. Indeed, if {Rfl} is an annihilating system in R, then {(Rfl}n} is ae. annihilating system in R n (here (Rfl)n is an ideal of Kn, consisting of all matrices with elements from R/3}. The set {H n (Rfl}n} bec'~mes, ~ter the repetitions, if any, are deleted, an annihilating system in H N Rn already.