Jordan*-derivations On Rings With Involution

The study of derivations,which are more active in operator algebra and operator theory,and has important theoretical and applied value.In recent decades,the research about all kinds of derivations has developed rapidly.Many resarch directions and research methods have been developed.And many fruitful results have been obtained.In this paper,we mainly discuss multiplicative and additive Jordan*-derivations on rings with an involution*and give some characterizations for them from different,aspects.The following are the main results in this paper.1.Let R be a*-ring with a.symmetric idempotent e.Assume that the characteristic of R is not 2,aRei = {0}(?)a = 0(i=1,2)and e2ae2x*e2 + e2xe2ae2 = 0(?)e2ae2 =0,where e1 = e,e2 = 1-e.Assume thatδ:R→R is a map.Ifδsatisfiesδ(ab + ba)=δ(a)b*+aδ(b)+δ(b)a*+ bδ(a)whenever ab = 0 for any a,b∈R,thenδ|eiRej is an additive Jordan*-derivation,whereδ|eiRej denotes the restriction to eiRej ofδand i,j∈{1,2}.Particularly,ifδis additive,thenδis a Jordan*-derivation,that is,δ(a2)=δ(a)a*+aδ(a)holds for all a∈R.2.Let R be a*-ring with a symmetric idempoteny e and unit 1.Assume that the characteristic of R is not 2,aRei = {0}(?)a = 0(i = 1,2)and e2ae2x*e2 +e2xe2ae2 =0(?)e2ae2 = 0,where e1 = e,e2 = 1-e.Then additive mapδ:R→R satisfiesδ(ab + ba)=δ(a)b*+ aδ(b)+δ(b)a*+ bδ(a)whenever ab + ba = 0 for any a,b∈R if and only ifδis a Jordan*-derivation.3.Let R be a*-ring with a symmetric idempotent e and unit 1.Assume that the characteristic of R neither 2 nor 3,1/2∈R,and aRei = {0}(?)a = 0(i = 1,2)with e1 = e,e2 = 1-e.If for any a∈R,there exists some integer n such that n1-a is invertible in R,then additive mapδ:R→R satisfiesδ(ab + ba)=δ(a)b*+ aδ(b)+δ(b)a*+ bδ(a)whenever ab = 1(ab + ba = 1)for any a,b∈R if and only if is a Jordan*-derivation.