We introduce the definition of Lie centralizers and investigate theadditivity of Lie centralizers on triangular rings. Characterizations of cen-tralizers and Lie centralizers on triangular rings and nest algebras are alsopresented

Jing, Wu

English

DigitalCommons@Fayetteville State University

Fayetteville State UniversityDigitalCommons@Fayetteville State UniversityMath and Computer Science Working Papers College of Arts and SciencesSpring 5-11-2011Additivity of Lie Centralizers on Triangular RingsWu JingFayetteville State University, wjing@uncfsu.eduFollow this and additional works at: http://digitalcommons.uncfsu.edu/macsc_wpThis Article is brought to you for free and open access by the College of Arts and Sciences at DigitalCommons@Fayetteville State University. It hasbeen accepted for inclusion in Math and Computer Science Working Papers by an authorized administrator of DigitalCommons@Fayetteville StateUniversity. For more information, please contact xpeng@uncfsu.edu.Recommended CitationJing, Wu, "Additivity of Lie Centralizers on Triangular Rings" (2011). Math and Computer Science Working Papers. Paper 8.http://digitalcommons.uncfsu.edu/macsc_wp/8ADDITIVITY OF LIE CENTRALIZERS ON TRIANGULARRINGSWU JINGAbstract. We introduce the definition of Lie centralizers and investigate theadditivity of Lie centralizers on triangular rings. Characterizations of cen-tralizers and Lie centralizers on triangular rings and nest algebras are alsopresented.1. Introduction and preliminariesLet R be a ring. An additive mapping T : R → R is called a left (resp. right)centralizer if T (ab) = T (a)b (resp. T (ab) = aT (b)) for any a, b ∈ R. T is called acentralizer if it is both a left and a right centralizer. Centralizers on rings as well asalgebras have been extensively investigated by many mathematicians (see [7], [8],[9], [10], [12], [14], and references therein). Motivated by the concept of centralizerson rings, we here introduce the definition of Lie centralizers as follows.Definition 1.1. Let R be a ring and δ : R → R an additive mapping. Then δ iscalled a Lie centralizer of R ifδ([a, b]) = [δ(a), b] (or δ([a, b]) = [a, δ(b)])holds true for any a, b ∈ R, where [a, b] = ab− ba is the usual Lie product of a andb.Remark 1.2. Observe that if δ([a, b]) = [δ(a), b], then we haveδ(ab− ba) = δ(a)b− bδ(a).Interchanging a and b in the above identity, we haveδ(ba− ab) = δ(b)a− aδ(b).Replacing a with −a in the above relation, we arrive at δ(ab− ba) = aδ(b)− δ(b)a,which can be written as δ([a, b]) = [a, δ(b)]. Thus conditions δ([a, b]) = [δ(a), b] andδ([a, b]) = [a, δ(b)] are equivalent regardless of the additivity of δ.Recall that an additive map of ring R into itself is called a commuting map if[T (a), a] = 0 for arbitrary a ∈ R.One can easily check that each centralizer is a Lie centralizer and every Liecentralizer is a commuting map.Over the last decades a lot of work has been done on the additivity of mappingson rings and operator algebras. We refer the readers to some recent papers [1, 5,6, 11, 15] where further references can be found. However, to author’s knowledge,2000 Mathematics Subject Classification. 16W25, 47L35.Key words and phrases. Lie centralizer; centralizer; triangular ring; nest algebra.The author is supported by Course Reassignment Program and HBCU Master’s Degree STEMProgram research mini-grant funded by Title III at Fayetteville State University.12 WU JINGthere has no result on additivity of commuting maps on rings. It is the aim of thispaper to initiate the study of additivity of commuting maps by investigating theadditivity of Lie centralizers on triangular rings. We will show that if δ is a map ofa triangular ring V = Tri(A,M,B) to itself satisfyingδ([A,B]) = [δ(A), B]for any A,B ∈ V, then δ = σ + τ , where σ : V → V is a centralizer and τ : V →Z(V) is a mapping such that τ(A+B) = τ(A)+τ(B)+ZA,B for some ZA,B ∈ Z(V)(depending on A and B) and τ([A,B]) = 0 for any A,B ∈ V. Characterizationsof centralizers and Lie centralizers on triangular rings and nest algebras are alsopresented.Recall that a triangular ring V = Tri(A,M,B) (see [2] and [13]) is a ring of theformV = Tri(A,M,B) ={[a m0 b]: a ∈ A,m ∈M, b ∈ B}under the usual matrix operations, where A and B are two rings over a commutativering R, andM is an (A,B)-bimodule which is faithful as a left A-module and alsoas a right B-module (see [13]). The center of V isZ(V) ={[a 00 b]: am = mb for all m ∈M}.Let piA : V→ A and piB : V→ B be the natural projections defined by[a m0 b]7→ a and[a m0 b]7→ brespectively. Then piA(Z(V)) ⊆ Z(A) and piB(Z(V)) ⊆ Z(B), and there exists aunique ring isomorphism ω : piA(Z(V)) → piB(V)) such that am = mω(a) for allm ∈M (see [11]).LetZ(A)V ={[a 00 0]: a ∈ Z(A)}andZ(B)V ={[0 00 b]: b ∈ Z(B)}.We setV11 ={[a 00 0]: a ∈ A},V12 ={[0 m0 0]: m ∈M},andV22 ={[0 00 b]: b ∈ B}.Then we may write V = V11⊕V12⊕V22, and every element A ∈ V can be writtenas A = A11 + A12 + A22. Note that notation Aij will denote an arbitrary elementof Vij .Let H be a Hilbert space over field F ∈ {R,C}. The algebra of all bounded linearoperators on H is denoted by B(H). For x, y ∈ H, we denote the inner product ofADDITIVITY OF LIE CENTRALIZERS ON TRIANGULAR RINGS 3these vectors by 〈x, y〉 and the rank one operator u 7→ 〈u, y〉x by x⊗ y. Note thatevery rank one operator on H can be written in this form.2. Additivity of Lie centralizers on triangular ringsIn this section, we aim to study the additivity of Lie centralizers on triangularrings. Note that, different from [2], [4], and [15], the rings A and B in triangularalgebra V = Tri(A,M,B) in most part of this section need not be unital. We alsowant to mention that our approaches are different from those in [2], [4], and [15]which mainly depend on the existence of identity elements in the underlying rings.In what follows, δ : V→ V will be a mapping such thatδ([A,B]) = [δ(A), B]holds true for all A,B ∈ V, where V = Tri(A,M,B) is a triangular ring satisfying(C1) if Am = {0} (resp. mB = {0}), then m = 0;(C2) piA(Z(V)) = Z(A) and piB(Z(V)) = Z(B).Lemma 2.1. For any A,B ∈ V, there exists a ZA,B ∈ Z(V) such thatδ(A+B) = δ(A) + δ(B) + ZA,B .Proof. For any A,B,X ∈ V, noticing that δ([A,B]) = [δ(A), B] = [A, δ(B)], wehave[δ(A+B), X] = [A+B, δ(X)]= [A, δ(X)] + [B, δ(X)]= [δ(A), X] + [δ(B), X]= [δ(A) + δ(B), X],which implies that δ(A + B) − δ(A) − δ(B) ∈ Z(R). Thus, there exists a ZA,B ∈Z(V) such thatδ(A+B) = δ(A) + δ(B) + ZA,B .Lemma 2.2. (1) δ(V11) ⊆ V11 ⊕Z(B)V.(2) δ(V22) ⊆ Z(A)V ⊕V22.Proof. We only prove (1). Let A11 ∈ V11 and write δ(A11) = U11 +U12 +U22. Forany B22 ∈ V22, we consider0 = δ([A11, B22])= [δ(A11), B22]= δ(A11)B22 −B22δ(A11)= U12B22 + U22B22 −B22U22,which yields that U12B22 = 0. Therefore, U12 = 0 and U22B22 = B22U22. Thus wecan conclude that δ(A11) = U11 + U22 with piB(U22) ∈ Z(B). By Lemma 2.2, we can define two mappings τ1 : V11 → Z(V) byτ1(A) = ω−1(piB([δ(A)]22))⊕ piB([δ(A)]22) for any A ∈ V11and τ2 : V22 → Z(V) byτ2(B) = piA([δ(B)]11)⊕ ω(piA([δ(B)]11)) for any B ∈ V22.4 WU JINGIt follows from the proof the above lemma that δ(A)−τ1(A) ∈ V11 for any A ∈ V11and δ(B)− τ2(B) ∈ V22 for any B ∈ V22.Lemma 2.3. δ(V12) ⊆ V12 ⊕Z(V).Proof. Pick A12 ∈ V12 and write δ(A12) = U11 + U12 + U22. For any X12 ∈ V12,we have0 = δ([A12, X12]) = [δ(A12), X12] = δ(A12)X12 −X12δ(A12) = U11X12 −X12U22.It follows that U11 + U22 ∈ Z(V). The above lemma enables us to define a mapping τ3 : V12 → Z(V) byτ3(A) = [δ(A)]11 + [δ(A)]22 for any A ∈ V12.It turns out, from the proof of Lemma 2.3, that δ(A)−τ3(A) ∈ V12 for any A ∈ V12.Suppose that τ3(A) = C1 ∈ Z(V) and τ3(A) = C2 ∈ Z(V) for A ∈ V12. ThenC2 − C1 = (δ(A)− C1)− (δ(A)− C2) ∈ V12 ∩ Z(V) = {0}.Thus, τ is well defined.We now continue to define a mapping τ : V→ Z(V) byτ(A) = τ1(A11) + τ2(A22) + τ3(A12)for any A = A11 +A12 +A22 ∈ V and a mapping σ : V→ V is then defined byσ(A) = δ(A)− τ(A).By the definitions of σ and τ , one can verify thatLemma 2.4. (1) σ(Vij) ⊆ Vij for 1 ≤ i ≤ j ≤ 2.(2) τ(A11 +A12 +A22) = τ(A11) + τ(A12) + τ(A22).Lemma 2.5. For any A,B ∈ V, by Lemma 2.1, there exists a ZA,B ∈ Z(V) suchthat δ(A+B) = δ(A) + δ(B) + ZA,B ∈ Z(V), then(1) τ(A+B) = τ(A) + τ(B) + ZA,B.(2) σ is additive.Proof. (1) In view of Lemma 2.4, we only need to consider the following cases:Case 1. τ(A12 +B12) = τ(A12) + τ(B12) + ZA12,B12 for some ZA12,B12 ∈ Z(V).It follows from Lemma 2.1 that there exists a ZA12,B12 ∈ Z(V) such thatδ(A12 +B12) = δ(A12) + δ(B12) + ZA12,B12 .Therefore,τ(A12 +B12)− τ(A12)− τ(B12)− ZA12,B12= δ(A12 +B12)− σ(A12 +B12)− δ(A12)+σ(A12)− δ(B12) + σ(B12)− ZA12,B12= σ(A12) + σ(B12)− σ(A12 +B12) ∈ Z(V) ∩V12 = {0}.Case 2. τ(Aii +Bii) = τ(Aii) + τ(Bii) + ZAii,Bii for some ZAii,Bii ∈ Z(V).With the same argument as in Case 1, we can getτ(Aii +Bii)− τ(Aii)− τ(Bii)− ZAii,Bii= σ(Aii) + σ(Bii)− σ(Aii +Bii) ∈ Z(V) ∩Vii = {0}.(2) It is a direct consequence of (1). ADDITIVITY OF LIE CENTRALIZERS ON TRIANGULAR RINGS 5Lemma 2.6. σ is a centralizer.Proof. We divide the proof into three steps.Step 1. σ(A11B12) = σ(A11)B12 and σ(A12B22) = σ(A12)B22.Indeed, by Lemma 2.4,σ(A11B12) = δ(A11B12) = δ([A11, B12])= [δ(A11), B12] = [σ(A11) + τ(A11), B12]= [σ(A11), B12] = σ(A11)B12 −B12σ(A11)= σ(A11)B12.One can check in a straightforward way that σ(A12B22) = σ(A12)B22.Step 2. σ(A11B11) = σ(A11)B11 and σ(A22B22) = σ(A22)B22.For arbitrary X12 ∈ V12, applying Step 1 twice, we haveσ(A11B11)X12 = σ(A11B11X12) = σ(A11)B11X12,which implies that [σ(A11B11) − σ(A11)B11]X12 = 0. Since M is a faithful leftA-module, we can conclude that σ(A11B11) = σ(A11)B11.Using the fact thatM is also a faithful right B-module, one can deduce σ(A22B22) =σ(A22)B22.Step 3. σ(AB) = σ(A)B.We write A = A11 + A12 + A22 and B = B11 + B12 + B22. One one hand, byLemma 2.5 and Steps 1 and 2 above, we haveσ(AB) = σ[(A11 +A12 +A22)(B11 +B12 +B22)]= σ(A11B11 +A11B12 +A12B22 +A22B22)= σ(A11B11) + σ(A11B12) + σ(A12B22) + σ(A22B22)= σ(A11)B11 + σ(A11)B12 + σ(A12)B22 + σ(A22)B22.On the other hand, applying Lemma 2.4, we getσ(A)B = σ(A11 +A12 +A22)(B11 +B12 +B22)= [σ(A11) + σ(A12) + σ(A22)](B11 +B12 +B22)= σ(A11)B11 + σ(A11)B12 + σ(A12)B22 + σ(A22)B22.Thus, σ is a left centralizer. In a similar manner, one can check that σ is also aright centralizer. Lemma 2.7. For any A,B ∈ V, τ([A,B]) = 0.Proof. Apply Lemma 2.6, we computeτ([A,B]) = δ([A,B])− σ([A,B])= [δ(A), B]− σ(AB −BA)= [σ(A) + τ(A), B]− σ(AB) + σ(BA)= [σ(A), B]− σ(A)B +Bσ(A)= 0.Based on the above lemmas, we can conclude that6 WU JINGTheorem 2.8. Let V = Tri(A,M,B) be a triangular ring satisfying(C1) if Am = {0} (resp. mB = {0}), then m = 0;(C2) piA(Z(V)) = Z(A) and piB(Z(V)) = Z(B).If mapping δ : V→ V has the property thatδ([A,B]) = [δ(A), B]for any A,B ∈ V, then δ = σ + τ , where σ : V → V is a centralizer and τ : V →Z(V) is a mapping such that for any A,B ∈ V(1) τ(A+B) = τ(A) + τ(B) + ZA,B for some ZA,B ∈ Z(V);(2) τ([A,B]) = 0.Particularly, we haveCorollary 2.9. Let V = Tri(A,M,B) be a triangular ring with the properties:(C1) if Am = {0} (resp. mB = {0}), then m = 0;(C2) piA(Z(V)) = Z(A) and piB(Z(V)) = Z(B).Then each Lie centralizer δ : V → V can be expressed as δ = σ + τ , where σ :V→ V is a centralizer and τ : V→ Z(V) is an additive mapping that vanishes atcommutators.We now turn our attention to centralizers on triangular rings. Different fromthe conditions in the above theorems, for a triangular ring W = Tri(A,M,B), weonly assume that both A and B are unital. The identity elements of A and B aredenoted by 1A and 1B respectively. We also use a⊕b and mˆ to denote the elements[a 00 b]and[0 m0 0]respectively. We take the liberty of borrowing Cheung’s method([2]) in the proof of the following theorem.Theorem 2.10. Let W = Tri(A,M,B) be a triangular ring, where both A and Bare unital. Suppose that mapping σ :W→W satisfiesσ(AB) = σ(A)B = Aσ(B)for all A,B ∈W, then σ is a centralizer and there exists a T ∈ Z(W) with piA(T ) ∈Z(A) and piB(T ) ∈ Z(B) such that for any A ∈Wσ(A) = TA.Proof. We writeσ([a m0 b])=[f11(a) + g11(m) + h11(b) f12(a) + g12(m) + h12(b)0 f22(a) + g22(m) + h22(b)],where f11 : A → A, f12 : A → M, f22 : A → B, g11 : M → A, g12 : M → M,g22 :M→ B, h11 : B → A, h12 : B →M, and h22 : B → B are maps.Note that a⊕ 0 = (a⊕ 0)(1A ⊕ 0). Then we compute σ(a⊕ 0) in two ways:[f11(a) f12(a)0 f22(a)]= σ(a⊕ 0) = σ[(a⊕ 0)(1A ⊕ 0)] = σ(a⊕ 0)(1A ⊕ 0)=[f11(a) f12(a)0 f22(a)] [1A 00 0]and[f11(a) f12(a)0 f22(a)]= σ(a⊕ 0) = σ[(a⊕ 0)(1A ⊕ 0)] = (a⊕ 0)σ(1A ⊕ 0)=[a 00 0] [f11(1A) f12(1A)0 f22(1A)].ADDITIVITY OF LIE CENTRALIZERS ON TRIANGULAR RINGS 7It follows that f12(a) = f22(a) = 0 and f11(a) = af11(1A).In a similar manner by using a⊕0 = (1A⊕0)(a⊕0), one can easily get f11(a) =f11(1A)a. Thus f11(1A) ∈ Z(A).With the same argument for 0 ⊕ b = (0 ⊕ b)(0 ⊕ 1B) = (0 ⊕ 1B)(0 ⊕ b) we candeduce that h11(b) = h12(b) = 0 and h22(b) = bh22(1B) = h22(1B)b for any b ∈ Band h22(1B) ∈ Z(B).Using the facts that mˆ = (1A ⊕ 0)mˆ and mˆ = mˆ(0 ⊕ 1B) we can infer thatg11(m) = g12(m) = 0 and g12(m) = f11(1A)m = mh22(1B) for any m ∈M.Therefore, σ takes the formσ([a m0 b])=[f11(1A)a g12(m)0 h22(1B)b],with f11(1A) ∈ Z(A), h22(1B) ∈ Z(B), and g12(m) = f11(1A)m = mh22(1B).LetT =[f11(1A) 00 h22(1B)],then T ∈ Z(W), piA(T ) ∈ Z(A), piB(T ) ∈ Z(B), andσ([a m0 b])= T[a m0 b].The proof of the theorem is thereby complete. Note that if both A and B are unital, Condition (1) in Theorem 2.8 is satisfied.Comparing with Theorem 2 in [2], we can have a sufficient condition for each Liecentralizer to be proper as follows.Corollary 2.11. Let W = Tri(A,M,B) be a triangular ring with both A and Bare unital. If piA(Z(W)) = Z(A) and piB(Z(W)) = Z(B), then each Lie centralizerδ onW is proper, that is, there is a T ∈ Z(W) and an additive map τ :W→ Z(W)vanishing at each commutator such thatδ(A) = TA+ τ(A)for all A ∈W.We complete this paper by considering the nest algebra case. Recall that a nestN in a Hilbert space H over the real or complex field F is a chain of orthogonalprojections on H including 0 and I which is closed in the strong operator topol-ogy. The nest algebra associated to N , denoted by AlgN , is the operator algebraconsisting of all bounded linear operators that leave N invariant, i.e.,AlgN = {A ∈ B(H) : AP = PAP for all P ∈ N}.Note that the center of AlgN is FI, where I is the identity operator on H.We first look at a very special case.Lemma 2.12. Let H be a Hilbert space H with dimH > 1. Suppose that mapσ : B(H)→ B(H) satisfiesσ(AB) = σ(A)B = Aσ(B)for arbitrary A,B ∈ B(H). Then σ is automatically linear and there is a λ ∈ Fsuch thatσ(A) = λAfor all A ∈ B(H).8 WU JINGProof. Since dimH > 1, we may pick vectors x0, y0 ∈ H with 〈x0, y0〉 = 1 anddefine a map T : H → H byTx = σ(x⊗ y0)x0for arbitrary x ∈ H.Claim 1. σ(A) = TA = AT for each A ∈ B(H).For any A ∈ B(H), we haveσ(Ax⊗ y0) = σ(A)x⊗ y0 = Aσ(x⊗ y0).Applying this equation to x0, we arrive atTAx = σ(A)x = ATxfor all x ∈ H. Consequently, TA = σ(A) = AT .Claim 2. T is linear.By the definition of T , for any x, y ∈ H and α, β ∈ F, we haveT (αx+ βy) = σ((αx+ βy)⊗ y0)x0= σ((αx+ βy)⊗ y0 · x0 ⊗ y0)x0= (αx+ βy)⊗ y0σ(x0 ⊗ y0)x0= αx⊗ y0σ(x0 ⊗ y0)x0 + βy ⊗ y0σ(x0 ⊗ y0)x0= ασ(x⊗ y0 · x0 ⊗ y0)x0 + βσ(y ⊗ y0 · x0 ⊗ y0)x0= ασ(x⊗ y0)x0 + βσ(y ⊗ y0)x0= αTx+ βTy.Claim 3. T is bounded.Let {xn} ⊆ H be an arbitrary sequence with xn → x and Txn → y as n → ∞.Using the fact that xn ⊗ y0 → x⊗ y0 as n→∞, we havey = limn→∞Txn= limn→∞σ(xn ⊗ y0)x0= limn→∞σ(xn ⊗ y0 · x0 ⊗ y0)x0= limn→∞xn ⊗ y0σ(x0 ⊗ y0)x0= x⊗ y0σ(x0 ⊗ y0)x0= σ(x⊗ y0 · x0 ⊗ y0)x0= σ(x⊗ y0)x0= Tx.By Closed Graph Theorem, we can infer that T is bounded.Claim 4. T = λI for some λ ∈ F.Note that the center of B(H) is FI. Now, it follows from Claims 1-3 that thereexists a scalar λ ∈ F such that T = λI.Theorem 2.13. Let AlgN be a nest algebra in a Hilbert space H with dimH > 1.Suppose that map σ : AlgN → AlgN satisfiesσ(AB) = σ(A)B = Aσ(B)ADDITIVITY OF LIE CENTRALIZERS ON TRIANGULAR RINGS 9for any A,B ∈ AlgN , then σ is of the form σ(A) = λA for some λ ∈ F. Moreover,σ is automatically linear.Proof. Case 1. N is a trivial nest; that is N = {0, I}. Then AlgN = B(H). Itfollows directly from Lemma 2.12.Case 2. N is nontrivial, i.e., there is P ∈ N\{0, I}. Then by [3], AlgN can beviewed as a triangular algebraAlgN =[Alg(PNP ) P (AlgN )(I − P )0 Alg((I − P )N (I − P ))].Thus, by Theorem 2.10, there exist a λ ∈ F such that σ(A) = λA for any A ∈ AlgNsince Z(AlgN ) = FI. Now, the automatic linearity of σ is obvious. In particular, we haveCorollary 2.14. Let AlgN be a nest algebra in a Hilbert space H over F withdimH > 1. If σ : AlgN → AlgH is a linear centralizer, then there exists a scalarλ ∈ F such that σ(A) = λA for all A ∈ AlgN .As for Lie centralizers on nest algebras, we have the following results.Corollary 2.15. Let AlgN be a nest algebra in a Hilbert space H with dimH > 1.Suppose that map δ : AlgN → AlgN satisfiesδ([A,B]) = [δ(A), B]for any A,B ∈ AlgN , then δ is of the form δ(A) = λA+ τ(A)I, where λ ∈ F andτ : AlgN → F is a mapping satisfying τ(A+ B) = τ(A) + τ(B) + λA,BI for someλA,B ∈ F and τ([A,B]) = 0 for any A,B ∈ AlgN .Corollary 2.16. Let AlgN be a nest algebra in a Hilbert space H over F withdimH > 1. If δ : AlgN → AlgN is a linear Lie centralizer, then δ(A) = λA +τ(A)I, where λ ∈ F and τ : AlgN → F is a linear mapping sending commutatorsto zero.References1. Z. Bai, S. Du, J. Hou, Multiplicative Lie isomorphisms between prime rings, Comm. Algebra36 (2008), 1626-1633.2. W. S. Cheung, Commuting maps of triangular algebras, J. London Math. Soc. 63 (2001),117-127.3. K. R. Davidson, Nest Algebras, Pitan Research Notes in Mathematics Series, Longman Sci-entific and Technical, 1988.4. P. Ji, W. Qi, Characterizations of Lie derivations of triangular algebras, Linear Algebra Appl.(in press) (doi:10.1016/j.laa.2011.02.048).5. W. Jing, F. Lu, Lie derivable mappings on prime rings, Linear Multilinear Algebra (to appear).6. W. Jing, F. Lu, Additivity of Jordan (triple) derivations on rings, Comm. Algebra (to appear).7. B. E. Johson, An introduction to the theory of centralizers, Proc. London Math. Soc. 14(1964), 299-320.8. P. Li, D. Han, W. Tang, Centralizers and Jordan derivations for CSL subalgebras of vonNeumann algebras, J. Operator Theorey (to appear).9. J. Vukman, Identities related to derivations and centralizers on standard operator algebras,Taiwanese J. Math. 11 (2007), 255-265.10. J. Vukman, I. Kosi-Ulbl, On centralizers of semiprime rings, Aequations Math. 66 (2003),277-283.11. Y. Wang, Additivity of multiplicative maps on triangular rings, Linear Algebra Appl. 434(2011), 625-635.10 WU JING12. Q. Wei, P. Li, Centralizers of J -subspace lattice algebras, Linear Algebra Appl. 426 (2007),228-237.13. T. Wong, Jordan isomorphism of triangular rings, Proc. Amer. Math. Soc. 133 (2005), 3381-3388.14. B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32 (1991),609-614.15. W. Yu, J. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432(2010), 2953-2960.Department of Mathematics & Computer Science, Fayetteville State University,Fayetteville, NC 28301E-mail address: wjing@uncfsu.edu

Additivity of Lie Centralizers on Triangular Rings

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Additivity of Lie Centralizers on Triangular Rings

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