Linear Group. Body - Generated By The Co-commutator Length

Let K be a division ring and ,n ≥ 2. Let res A = rank(A-In)for any A ∈ GLn(K). 1-involution is a matrix M ∈ SLn(K) similar to (-1) In-1 with ChK 2.S-dilatation B ∈ SLn(K) is a matrix similar to (a) In-1(a≠1). Matrix A 6 SLn(K) is a quasi-dilatation,if the residue matrix of A is a quasi-central matrix, when res A ≥ 2.We prove that when ChK≠ 2,K≠ F3, every martix A in SLn(K) is a product of at most [resA/2] + 1 commutators of 1-involution or [resA/2] + 1 commutators of 1-involution with a S-dilatation, if A is not a quasi-dilatation .Otherwise,A is a product of at most [resA/2] + 2 commutators of 1-involution or [resA/2] + 2 commutators of 1-involution with a S-dilatation.From this basic result we easily deduce,in Theorem 3.4,the element of GLn(K) is a product of commutators of 1-involutation with dilatation,and determine the smallest length of this factorization.