| Super algebras first appeared in physics, in the Theory of Supersymmetry, to create an algebraic structure representing the behavior of the subatomic particles known as bosons and fermions. More and more authors are interested in superalgebras, which promote the investigation of physics and other subjects.Functional identities (FI) theory is a relatively new theory. The goal of the general FI theory is to determine the form of these functions or to determine the structure of the ring admitting the FI in question. It originated from the results of commuting mappings on rings. FI theory is applicable to different areas, such as nonassociative algebras, linear algebras, operator theory, and mathematical physics.In1961, Herstein conjectured that Lie isomorphisms (Lie derivations) of Lie subalge-bras of some associative algebras are induced by isomorphisms (derivations)? Martindale and his school continued Herstein’s investigation. They investigated Lie mappings on rings with non-trivial idempotents. In the process Martindale constructed the Martin-dale quotient ring and the extended centroid, which play an important role in the theory of rings.In1957, Posner proved that the existence of a nonzero centralizing derivation on a prime ring R implies that R is commutative. He initiated the study of commut- ing mappings. Authors studied commuting mappings in different ways from then on. Bresar described the commuting additive mappings of prime rings. It plays an impor-tant role to study commuting mappings and it is a result on functional identities. Later, Bresar investigated commuting biadditive mappings. As an application, the structures of commutativity-preserving mappings, Lie isomorphisms and Lie derivations of prime rings were determined. Subsequently, Bresar investigated FI and GFI of degree two in prime rings. Chebotar studied GFI of arbitrary degree. The crucial tool in the theory is the d-free set, which was developed by Beidar and Chebotar in2000. As an application, Beidar and other scholars gave the solutions of all the Herstein’s problems concerning Lie type. They also described Lie admissible mappings and mappings preserving algebraic properties of elements. In the paper, we describe Lie superderivations of superalgebras, Jordan superderivations of prime superalgebras and mappings of Poisson superalgebras, using the theory of functional identities in superalgebras.Theorem2.2.1Let A be any superalgebra, and let Q be a unital superalgebra with center C. Suppose that d:A'Q is a Lie superderivation. If A is a4-superfree subset of Q, then there exist a superderivation f:A'Q and a linear mapping h:A'C+Cω such that d(x)=f(x)+h(x) for all x∈A.Theorem2.3.1Let A=A0(?) A1be a prime superalgebra with maximal right ring of quotients Q and extended centroid C. Suppose that a:A'Q is a Jordan superderivation. If deg(A1)≥9, then a:A'Q is a superderivation.Theorem2.4.1Let A be a Poisson superalgebra and a:A'Q be a mapping of degree k, k=0,1, such that {x,y}α=[xα,yα]s for all x,y∈A. If Aα is a4-superfree subset of Q. Then there exist λij,ηij∈C+Cω, μij:Ai'C+Cω,ζij:Aj'C+Cω and v:A×A'C+Cω, i, j∈{0,1}, such that for all x,y∈A. A prime superalgebra A is a Z2-graded algebra and A is semiprime as an algebra. One can construct the extended centroid C=C0(?) C1of A, and all nonzero homoge-neous elements in C are invertible. Fosner investigated the extended centroid of prime superalgebras. As an application, he generalized three well-known results of prime rings to prime superalgebras:the theorem of Posner on the product of two derivations, and Bresar’s descriptions of biderivations and commuting additive mappings. In the paper, we describe generalized superderivations on a prime superalgebra. Using the result we study the produce of two generalized superderivations and a linear combination of generalized superderivations on prime superalgebras.Theorem3.1.1Let A be a prime superalgebra and g:A'A be a generalized superderivation. Then g can be extended to Q and there exist an element a∈Q and a superderivation d of A such that g(x)=ax+d(x) for all x∈A, where both a and d are determined by g uniquely.Theorem3.2.1Let A be a prime superalgebra and let f＝a+d and g=b+k be two nonzero generalized superderivations on A, where a,b■Q and both d and k are superderivations on A. If fg is also a generalized superderivation on A. Then one of the following cases is true:(ⅰ) There exists0≠ω∈Co such that ukj(x)+dj(x)=0;(ⅱ)[ai,x]s+di(x)=0;(in)[bi, x]s+ki(x)=0for all x∈A, where i,j∈{0,1}, ai,bi∈Qi and both di and ki are superderivations of degree i on A, as well as dj and kj.Theorem3.3.1Let A be a prime superalgebra. Let d,f, g, h be nonzero generalized superderivations of A and ω, α, β,γ be nonzero superderivations associated with d,f,g, h. If there exist a,b,c∈A such that d(x)=af{x)+bg(x)+ch(x) for all x∈A, then(ⅰ)ω(x)=aa(x)+bβ(x)+cγ(x) for all x∈A; (ii) Either one of{a,b,1},{b, c,1},{a, c,1} is a C-dependent set or there exist λ,ρ,ω∈C not all zero and such that λf+ρg+εh is a left multiplication mapping.Let R be a commutative ring. Let A and B be unital algebras over R and M be a (A, B)-bimodule which is faithful as a left A-module as well as a right B-module. The algebra under the usual matrix operations is said to be a triangular algebra. Basic examples of tri-angular algebras are upper triangular matrix algebras and nest algebras. In2001, Cheung described commuting linear mappings of triangular algebras. A number of authors in-vestigated Lie mappings, Jordan mappings, multiplicative derivations, and k-commuting mappings on triangular algebras from then on.Let R be a ring. A mapping f:R'R is called strong commutativity preserving on R if [f(x),f(y)]=[x,y] for all x, y∈R. In1994, Bresar and Miers investigated strong commutativity preserving additive mappings on semiprime rings. They proved that if f,g:R'R are additive mappings of a semiprime ring R such that f is onto and [f(x),g(x)]=[x,y] for all x,y∈R, then there exist an invertible element λ∈C and additive mappings ξ,η:R'C such that f{x)=λx+ξ(x) and g(x)=λ-1x+η(x) for all x∈R. In the paper, we investigate strong commutativity preserving generalized derivations ([g1(x),g2(y)]=[x,y]) on triangular algebras. As consequences strong com-mutativity preserving generalized derivations ([g1(x),g2(y)]=[x,y]) on upper triangular matrix algebras and nest algebras are determined.Theorem4.1.1Let U be a triangular algebra. Let g be a generalized derivation of U. Then for all a∈A,b∈B,m∈M, where α0∈A,b0∈B, s, t∈M and(i) PA is a derivation of A, f(am)=PA(α)m+αf(m);(ii) PB is a derivation of B, f(mb)=mPB(b)+f(m)b.Theorem4.2.1Let U=Tri(A, M, B) be a triangular algebra such that either A or B has no nonzero central ideals. If g1,g2are generalized derivations such that [g1(x),g2(y)]=[x,y] for all x,y∈U,then g1(x)=λ-1x+[x,u] and g2(x)=λ2g1(x) for all x∈U,where λ∈Z(U)and u∈U with u[U,U]=0=[U,U]u.Theorem4.3.1Let Tn(S) be an upper triangular matrix algebra with n≥3.If g1,g2are generalized derivations of Tn(S) such that [g1(x),g2(y)]=[x,y] for all x,y∈Tn(S),then there exist λ∈Z(Tn(S))and A∈Tn(S) with the property A[Tn(S),Tn(S]=0=[Tn(S),Tn(S)]A such that g1(x)=λ-1x+[x,A] and g2(x)=λ2g1(x)for all x∈Tn(S).Theorem4.4.1Let N6e a nest of a complex Hilbert space H with dim(H)>2.If g1,g2are generalized derivations of T(N) such that [g1(x),g2(y)]=[x,y] for all x,y∈T(N),then there exist λ∈C and A∈T(N)with the propetry A[T(N),T(N)]=0=[T(N),T(N)]A such that g1(x)=λ-1x+[x,A] and g2(x)=λ1g1(x) for all x∈T(N). |
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