A Map That Strongly Preserves The K-Jordan Product On The Ring

Let R be a ring and k be any positive integer.For any x,y ∈ R,the k-Jordan product of x,y is defined by {x,y}k = {{x,y}k-1,y}1,where {x,y}0 = x,{x,y}1 =xy+yx.It is clear that k-Jordan product is the usual Jordan product if k= 1.Assume that f:R→R is a map.f is strong k-Jordan product preserving if {f(x),f(y)}k = {x,y}k holds for all x,y ∈ R.In this thesis,we mainly discuss the structures of strong k-Jordan product preserving maps on some rings.The following are the main results in this paper.1.Assume that R is a unital ring with a nontrivial idempotent e and characteristic not 2.If R satisfies aRe = {0} a = 0 and aR(1-e)= {0}(?)a = 0,then a surjective map f:R → R is strong k-Jordan product preserving if and only if there exists a λ ∈L(R)(the center of R)with λk+1=1 such that f(x)= x holds for all x ∈ R.2.Assume that R is a unital prime ring with a nontrivial idempotent and charac-teristic 2.Then a surjective map f:R→R is strong 2-Jordan product preserving if and only if there exists some λ ∈ C(the extended centroid of R)with λ 3 = 1 such that f(x)= λx + μ(x)holds for all x ∈R,where μ:R→C is a map.3.Let A and B be two unital rings and let M be a faithfull(A,B)-bimodule.Assume that u=Tri(A,M,B)is the triangular ring with characteristic not 2.Assume thatΦ:u→u is a map.The following results hold.(1)If Φ is surjective,then Φ is strong k-Jordan product preserving if only if Φ(Ⅹ)=Φ(Ⅰ)X holds for all X ∈ u,where Φ(Ⅰ)is in the center of u with Φ(Ⅰ)k+1=Ⅰ(2)If Φ is additive,then Φ is strong 2-Jordan product preserving if only if Φ(Ⅹ)=S(Ⅰ)X holds for all X ∈ u,where Φ(Ⅰ)is in the center of u with Φ(Ⅰ)3 = Ⅰ.