?-skew Lie Product On Rings With Involution

Let R be a ring with an involution*and k be a positive integer.For any a,b ?R,the k-skew Lie product of a,b is defined by*[a,b]k?[a,*(a,b]k-1]1,where*[a,b]0=b,*[a,b]1 = ab-ba*.Assume that f:R?R is an additive map.f is k-skew commuting if*[a,f(a)]k= 0 holds for all a ?R.In this thesis,we mainly discuss the properties and structures of ?-skew Lie product and ?-skew commuting additive maps.The following are the main results.1.If R is an either noncommutative prime*-ring or commutative prime*-ring with a nonsymmetric element,then s ?R satisfies*[a,s]k = 0 for all a ? R must imply s? 0.2.If R is a unital prime*-ring with characteristic not 2 and satisfies the following two conditions:(i)R contains a nontrivial symmetric idempotent e,(ii)either eRe or(1-e)R(1-e)is noncommutative,then f(x)= 0 for all x ? R.3.If R,is 2-torsion free,unital,contains a nontrivial symmetric idempotent e1,and satisfies the following two conditions:(i)for any a ? R,aRei = {0} implies a = 0,where i = 1,2,e2 = 1-e1;(ii)for s ? R,*eiaei,eisei]k ?Z(eiRei)holds for any a ?R,awhich implies eisei ? 0(i= 1,2).Let f:R? R be additive maps,k? 1 any positive integer,then for any a ? R,we have f(a)= 0.