BRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR.

REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 60, No. 1, 2019, Pages 9 20 Published online: February 11, 2019 https://doi.org/10.33044/revuma.v60n1a02 LIE n-multiplicative MAPPINGS ON TRIANGULAR n-matrix RINGS BRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR. Abstract. We extend to triangular n-matrix rings and Lie n-multiplicative maps a result about Lie multiplicative maps on triangular algebras due to Xiaofei Qi and Jinchuan Hou. 1. Introduction Let R be an associative ring and [x 1, x 2 ] = x 1 x 2 x 2 x 1 denote the usual Lie product of x 1 and x 2. Let us define the following sequence of polynomials: p 1 (x) = x and p n (x 1, x 2,..., x n ) = [p n 1 (x 1, x 2,..., x n 1 ), x n ] for all integers n 2. Thus, p 2 (x 1, x 2 ) = [x 1, x 2 ], p 3 (x 1, x 2, x 3 ) = [[x 1, x 2 ], x 3 ], etc. Let n 2 be an integer. Assume that S is any ring. A map ϕ : R S is called a Lie n-multiplicative mapping if ϕ(p n (x 1, x 2,..., x n )) = p n (ϕ(x 1 ), ϕ(x 2 ),..., ϕ(x n )). (1) In particular, if n = 2, ϕ will be called a Lie multiplicative mapping. And, if n = 3, ϕ will be called a Lie triple multiplicative mapping. The study on the question of when a particular application between two rings is additive has become an area of great interest in the theory of rings. One of the first results ever recorded was given by Martindale III, which in his condition requires that the ring possess idempotents, see [4]. Xiaofei Qi and Jinchuan Hou [6] also considered this question in the context of triangular algebras. They proved the following theorem. Theorem 1.1 (Xiaofei Qi and Jinchuan Hou [6]). Let A and B be unital algebras over a commutative ring R, and let M be a (A, B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri(A, M, B) be the triangular algebra and V any algebra over R. Assume that Φ : U V is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST T S) = Φ(S)Φ(T ) Φ(T )Φ(S) S, T U. Then Φ(S + T ) = Φ(S) + Φ(T ) + Z S,T for all S, T U, where Z S,T is an element in the centre Z(V) of V depending on S and T. 2010 Mathematics Subject Classification. 47L35; 16W25. Key words and phrases. Triangular n-matrix rings; Additivity; Lie n-multiplicative maps. 9

10 BRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR. This motivated us to discuss the additivity of Lie n-multiplicative mappings on another kind of rings: triangular n-matrix rings. In this paper, we give a full answer to this question, where Xiaofei Qi and Jinchuan Hou s result is a consequence of our case. 2. Motivation and definitions For any unital ring R, let Mod(R) denote the category of unitary R-modules, i.e., satisfying 1m = m for all elements m. This category is important in many areas of mathematics such as ring theory, representation theory, and homological algebra. The purpose of this paper is to work with a more general category, that is, Mod(R) where R does not have identity element. In this paper R denotes the unital ring R Z. Define operations on R by (r, λ) + (t, µ) := (r + t, λ + µ), (r, λ) (t, µ) := (rt + λt + µr, λµ). Then R is a ring with (0 R, 1) := 1 R as multiplicative identity. If M is a non unitary R-module, define a right R-module operation by and a left R-module operation by (r, λ)m := rm + λm m(r, λ) := mr + λm, where the action Z on M is the usual of M as a Z-module. A module over R is the same thing as a unitary R-module. The following definition is a generalization of the definition that arises in the work of W. S. Cheung [1]. This definition appears in Ferreira s paper [3]. Definition 2.1. Let R 1, R 2,..., R n be rings and M ij (R i, R j )-bimodules with M ii = R i for all 1 i j n. Let ϕ ijk : M ij Rj M jk M ik be (R i, R k )- bimodules homomorphisms with ϕ iij : R i Ri M ij M ij and ϕ ijj : M ij Rj R j M ij the canonical multiplication maps for all 1 i j k n. Write ab = ϕ ijk (a b) for a M ij, b M jk. We consider Let (i) M ij is faithful as a left R i -module and faithful as a right R j -module i < j. (ii) if m ij M ij is such that R i m ij R j = 0 then m ij = 0 i < j. T = r 11 m 12... m 1n r 22... m 2n.... r nn n n : r ii R i (= M ii ), m ij M ij }{{} (1 i<j n) be the set of all n n matrices [m ij ] with the (i, j)-entry m ij M ij for all 1 i j n. Observe that, with the obvious matrix operations of addition and multiplication, T is a ring iff a(bc) = (ab)c for all a M ik, b M kl and c M lj

LIE n-multiplicative MAPPINGS ON TRIANGULAR n-matrix RINGS 11 for all 1 i k l j n. When T is a ring, it is called a triangular n-matrix ring. Note that if n = 2 we have the triangular matrix ring. As in [3] we denote by r 11 0... 0 n r r 22... 0 ii the element... in T.. r nn { { } m ij, if (k, t) = (i, j) Set T ij = (m kt ) : m kt = 0, if (k, t) (i, j), i j. Then we can write T = T ij. Henceforth the element a ij belongs to T ij and the corresponding elements are in R 1,..., R n or M ij. By a direct calculation a ij a kl = 0 if j k. Also as in [3] we define natural projections π Ri : T R i (1 i n) by r 11 m 12... m 1n r 22... m 2n.... r ii. r nn Definition 2.2. Let R, S be rings; we shall say that the Lie n-multiplicative mapping ϕ : R S is almost additive if there exist S A,B in the centre Z(S) of S depending on A and B such that for all A, B R. ϕ(a + B) = ϕ(a) + ϕ(b) + S A,B The proposition below appears in [3]; it is a generalization of Proposition 3 of [1] and will be very useful. Proposition 2.1. Let T be a triangular n-matrix ring. The center of T is { n } Z(T) = r ii r ii m ij = m ij r jj for all m ij M ij, i < j. Furthermore, Z(T) ii = πri (Z(T)) Z(R i ), and there exists a unique ring isomorphism τ j i from π Ri (Z(T)) to π Rj (Z(T)) i j such that r ii m ij = m ij τ j i (r ii) for all m ij M ij. Remark 2.1. Throughout this paper we shall make some identifications. example: let r kk R k and m ij M ij ; then 0 0...... 0... 0....... 0... 0 0 r kk r kk 0 0 0 0.... 0 For

12 BRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR. and 0 0...... 0... 0....... 0 m ij 0 m ij..... 0 0.... 0 where i, j, k {1, 2,..., n}. In addition we have the following identifications: Let R = R Z be a unital ring. If r R then r (r, 0). And ϕ Id : R Z S Z with (ϕ Id)(r, λ) = (ϕ(r), Id(λ)) = (ϕ(r), λ), where Id is the identity map on Z. By a straightforward calculus it is shown that ϕ Id is a Lie n-multiplicative mapping. Sometimes we shall do ϕ Id ϕ. 3. A key lemma The following results are generalizations of those that appear in [6]. Lemma 3.1 (Standard lemma). Let A, B, C R and ϕ(c) = ϕ(a) + ϕ(b). Then for any T 1, T 2,..., T n 1 R, we have ϕ(p n (C, T 1, T 2,..., T n 1 )) = ϕ(p n (A, T 1, T 2,..., T n 1 )) Proof. Using (1) we have + ϕ(p n (B, T 1, T 2,..., T n 1 )). ϕ(p n (C, T 1, T 2,..., T n 1 )) = p n (ϕ(c), ϕ(t 1 ), ϕ(t 2 ),..., ϕ(t n 1 )) Note that if = p n (ϕ(a) + ϕ(b), ϕ(t 1 ), ϕ(t 2 ),..., ϕ(t n 1 )) = p n (ϕ(a), ϕ(t 1 ), ϕ(t 2 ),..., ϕ(t n 1 )) + p n (ϕ(b), ϕ(t 1 ), ϕ(t 2 ),..., ϕ(t n 1 )) = ϕ(p n (A, T 1, T 2,..., T n 1 )) ϕ(p n (A, T 1, T 2,..., T n 1 )) + ϕ(p n (B, T 1, T 2,..., T n 1 )) then by the injectivity of ϕ we get + ϕ(p n (B, T 1, T 2,..., T n 1 )). = ϕ(p n (A, T 1, T 2,..., T n 1 ) + p n (B, T 1, T 2,..., T n 1 )), p n (C, T 1, T 2,..., T n 1 ) = p n (A, T 1, T 2,..., T n 1 ) + p n (B, T 1, T 2,..., T n 1 ). Lemma 3.2. Let R 1, R 2,..., R n be rings and M ij (R i, R j )-bimodules as in Definition 2.1. Let T be the triangular n-matrix ring. If p 2 (A, T) Z(T) then A Z(T) for each A T. Moreover, if p n (A, T,..., T) Z(T) then A Z(T) for each A T.

LIE n-multiplicative MAPPINGS ON TRIANGULAR n-matrix RINGS 13 Proof. Let A = a ij such that p 2 (A, T) Z(T). Thus p 2 (A, R k ) Z(T) for k = 1,..., n 1, and it follows that p 2 (A, R k ) = i p 2 (a ik, R k ) + j p 2 (a kj, R k ) + p 2 (a kk, R k ), because If i < j, k i and k j then p 2 (a ij, R k ) = 0; If i = j and k i then p 2 (a ii, R k ) = 0. As p 2 (A, R k ) Z(T) we have p 2 (A, R k ) = p 2 (a kk, R k ) for k = 1,..., n 1 by Proposition 2.1. Consequently, p 2 (A, R k )m kj = 0 for k < j. Since M kj is faithful as a left R k -module, we see that p 2 (A, R k ) = 0 for k = 1,..., n 1. In the case k = n we use an analogous argument and we obtain m in p 2 (A, R n ) = 0. And since M in is faithful as a right R n -module, it follows that p 2 (A, R n ) = 0. Now note that p 2 (A, M ij ) = p 2 (A ii, M ij ) + p 2 (A jj, M ij ) and as p 2 (A, M ij ) Z(T) we get p 2 (A, M ij ) = 0 for i < j. Therefore p 2 (A, T) = 0, that is, A Z(T). 4. Main results Let s state our main result in this section, which is a generalization of Theorem 2.1 in [6]. Theorem 4.1. Let T be the triangular n-matrix ring and S any ring. Consider ϕ : T S a bijection Lie n-multiplicative mapping satisfying Then (i) ϕ(z(t)) Z(S). ϕ(a + B) = ϕ(a) + ϕ(b) + S A,B for all A, B T, where S A,B is an element in the centre Z(S) of S depending on A and B. To prove Theorem 4.1 we introduce a set of lemmas, almost all of which are generalizations of claims in [6]. We begin with the following lemma. Lemma 4.1. ϕ(0) = 0. Proof. Indeed, ϕ(0) = ϕ(p n (0, 0,..., 0)) = p n (ϕ(0), ϕ(0),..., ϕ(0)) = [ [[ϕ(0), ϕ(0)], ϕ(0)],... ] = 0. Lemma 4.2. For any A T and any Z Z(T), there exists S Z(S) such that ϕ(a + Z) = ϕ(a) + S. Proof. Note that ϕ 1 is also a bijection Lie n-multiplicative map. Let A T, Z Z(T) and T 2,..., T n T. As ϕ 1 is surjective we have ϕ 1 (S ) = A and

14 BRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR. ϕ 1 (S) = Z. Now as ϕ(z(t)) Z(S) by condition (i) of Theorem 4.1 we have ϕ(p n (ϕ 1 (S + S), T 2,..., T n )) = p n (S + S, ϕ(t 2 ),..., ϕ(t n )) = p n (S, ϕ(t 2 ),..., ϕ(t n )) = ϕ(p n (ϕ 1 (S ), T 2,..., T n )). Thus, p n (ϕ 1 (S + S) ϕ 1 (S ), T 2,..., T n ) = 0, and it follows that ϕ 1 (S + S) ϕ 1 (S ) Z(T) by Lemma 3.2. Therefore ϕ(a + Z) = ϕ(a) + S. Lemma 4.3. For any a kk R k, a ii R i, a jj R j and m ij M ij, i < j, there exist S, S 1, S 2 Z(S) such that ϕ(a kk + m ij ) = ϕ(a kk ) + ϕ( m ij ) + S; i<j i<j ϕ(a ii + m ij ) = ϕ(a ii ) + ϕ(m ij ) + S 1 ; ϕ(a jj + m ij ) = ϕ(a jj ) + ϕ(m ij ) + S 2. Proof. We shall only prove the first item because the demonstrations of the others are similar. It is important to note that if ϕ is Lie 2-multiplicative mapping (i.e., Lie multiplicative mapping), then ϕ is also Lie n-multiplicative mapping for any n 3. Therefore, the following proof is valid for all n 2. As ϕ is surjective, there is an element H = h ij T such that ϕ(h) = ϕ(a kk ) + ϕ( i<j m ij ). Let b jj R j, j k and T 3,..., T n T; by Lemma 3.1 we have ϕ(p n (H, b jj, T 3,..., T n )) = ϕ(p n (a kk, b jj, T 3,..., T n )) + ϕ(p n ( i<j m ij, b jj, T 3,..., T n )) = ϕ(p n (m ij, b jj, T 3,..., T n )). It follows that p n (H m ij, b jj, T 3,..., T n ) Z(T) and by Lemma 3.2 we get p 2 (H m ij, b jj ) Z(T), thus p 2 (h jj, b jj ) Z(T). Therefore p 2 (h ij m ij, b jj ) = 0, i<j that is (h ij m ij )b jj = 0 for all b jj R j and by condition (ii) of Definition 2.1 we get h ij = m ij. Now consider b kj M kj ; by standard Lemma 3.1 we have ϕ(p n (H, b kj, T 3,..., T n )) = ϕ(p n (a kk, b kj, T 3,..., T n )) + ϕ(p n ( i<j m ij, b kj, T 3,..., T n )) = ϕ(p n (a kk, b kj, T 3,..., T n )) + ϕ(0) = ϕ(p n (a kk, b kj, T 3,..., T n )). It follows that p n (H a kk, b kj, T 3,..., T n ) = 0 and by Lemma 3.2 we have p 2 (H a kk, b kj ) Z(T). Thus by Proposition 2.1 we get p 2 (H a kk, b kj ) = 0, which implies that (h kk a kk )b kj = b kj h jj for all b kj M kj. Therefore by Proposition 2.1

LIE n-multiplicative MAPPINGS ON TRIANGULAR n-matrix RINGS 15 we obtain l {k,j} verify that the lemma is valid. h ll + (h kk a kk + h jj ) Z(T). And finally by Lemma 4.2 we Lemma 4.4. For any m ij, s ij M ij with i < j, we have ϕ(m ij + s ij ) = ϕ(m ij ) + ϕ(s ij ). Proof. Firstly we note that for any m ij, s ij M ij, i < j, the following identity is valid: m ij + s ij = p n (1 Ri + s ij, 1 Rj + m ij, 1 Rj,..., 1 Rj ). Indeed, due to Remark 2.1 we get m ij + s ij = p 2 (1 Ri + s ij, 1 Rj + m ij ). It follows that m ij + s ij = p n (1 Ri + s ij, 1 Rj + m ij, 1 Rj,..., 1 Rj ). Finally by Lemma 4.3 we have ϕ(m ij + s ij ) = ϕ(p n (1 Ri + s ij, 1 Rj + m ij, 1 Rj,..., 1 Rj )) = p n (ϕ(1 Ri + s ij ), ϕ(1 Rj + m ij ), ϕ(1 Rj ),..., ϕ(1 Rj )) = p n (ϕ(1 Ri ) + ϕ(s ij ) + S 1, ϕ(1 Rj ) + ϕ(m ij ) + S 2, ϕ(1 Rj ),..., ϕ(1 Rj )) = p n (ϕ(1 Ri ) + ϕ(s ij ), ϕ(1 Rj ) + ϕ(m ij ), ϕ(1 Rj ),..., ϕ(1 Rj )) = p n (ϕ(1 Ri ), ϕ(1 Rj ), ϕ(1 Rj ),..., ϕ(1 Rj )) + p n (ϕ(1 Ri ), ϕ(m ij ), ϕ(1 Rj ),..., ϕ(1 Rj )) + p n (ϕ(s ij ), ϕ(1 Rj ), ϕ(1 Rj ),..., ϕ(1 Rj )) + p n (ϕ(s ij ), ϕ(m ij ), ϕ(1 Rj ),..., ϕ(1 Rj )) = ϕ(p n (1 Ri, 1 Rj, 1 Rj,..., 1 Rj )) + ϕ(p n (1 Ri, m ij, 1 Rj,..., 1 Rj )) + ϕ(p n (s ij, 1 Rj, 1 Rj,..., 1 Rj )) + ϕ(p n (s ij, m ij, 1 Rj,..., 1 Rj )) = ϕ(p n (1 Ri, m ij, 1 Rj,..., 1 Rj )) + ϕ(p n (s ij, 1 Rj, 1 Rj,..., 1 Rj )) = ϕ(0) + ϕ(m ij ) + ϕ(s ij ) + ϕ(0) = ϕ(m ij ) + ϕ(s ij ). Lemma 4.5. For any a ii, b ii R i, i = 1, 2,..., n, there exists S i Z(S) such that ϕ(a ii + b ii ) = ϕ(a ii ) + ϕ(b ii ) + S i. Proof. As ϕ is surjective, there is an element H = h ij T such that ϕ(h) = ϕ(a ii ) + ϕ(b ii ). Let c kk R k, k i and T 3,..., T n T; by Lemma 3.1 we have ϕ(p n (H, c kk, T 3,..., T n )) = ϕ(p n (a ii, c kk, T 3,..., T n )) + ϕ(p n (b ii, c kk, T 3,..., T n )) = ϕ(0) + ϕ(0) = 0 It follows that p n (H, c kk, T 3,..., T n ) Z(T) and by Lemma 3.2 we get p 2 (H, c kk ) Z(T), thus p 2 (h kk, c kk ) Z(T). Therefore by condition (i) of Definition 2.1 we

16 BRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR. have h kk Z(R k ). Moreover, i<j p 2 (h ij, c kk ) = 0, that is, h ik c kk = 0 for all c kk R k and by condition (ii) of Definition 2.1 we get h ik = 0. Now consider c ij M ij and r jj R j ; by standard Lemma 3.1 and Lemma 4.4 we have ϕ(p n (H, c ij, r jj,..., r jj )) = ϕ(p n (a ii, c ij, r jj,..., r jj )) + ϕ(p n (b ii, c ij, r jj,..., r jj )) = ϕ(p n (a ii + b ii, c ij, r jj,..., r jj )). It follows that p n (H (a ii + b ii ), c ij, r jj,..., r jj ) = 0 and by (ii) of Definition 2.1 we get (h ii (a ii + b ii ))c ij = c ij h jj for all c ij M ij. Therefore by Proposition 2.1 we obtain h ll + h ii (a ii + b ii ) + h jj Z(T). And finally by Lemma 4.2 we l {i,j} see that the Lemma is valid. Lemma 4.6. For any T T with T = that ϕ(t ) = ϕ(t ij ) + S. Proof. As ϕ is surjective, there is an element H = ϕ(h) = n ϕ(t tt ) + t=1 Let c kk R k, k = 1, 2,..., n; by Lemma 3.1 we have T ij, there exists S Z(S) such ϕ(t ij ). ϕ(p n (H, c kk, c kk,..., c kk )) = ϕ(p n (T kk, c kk, c kk,..., c kk )) k 1 + + H ij T such that ϕ(p n (T ik, c kk, c kk,..., c kk )) n j=k+1 ϕ(p n (T kj, c kk, c kk,..., c kk )). Now let c ll R l with l {1,..., k 1}; again by Lemma 3.1 we get ϕ(p n (p n (H, c kk, c kk,..., c kk ), c ll, c ll,..., c ll )) Since ϕ is injective we have p n (p n (H, c kk, c kk,..., c kk ), c ll, c ll,..., c ll ) = ϕ(p n (p n (T lk, c kk, c kk,..., c kk ), c ll, c ll,..., c ll )). = p n (p n (T lk, c kk, c kk,..., c kk ), c ll, c ll,..., c ll ). It follows that p n (p n (H T lk, c kk, c kk,..., c kk ), c ll, c ll,..., c ll ) = 0 and by (ii) of Definition 2.1 we obtain H lk = T lk. Again let c qq R q with q {k + 1,..., n}; by

LIE n-multiplicative MAPPINGS ON TRIANGULAR n-matrix RINGS 17 Lemma 3.1 we get ϕ(p n (p n (H, c kk, c kk,..., c kk ), c qq, c qq,..., c qq )) Since ϕ is injective we have p n (p n (H, c kk, c kk,..., c kk ), c qq, c qq,..., c qq ) = ϕ(p n (p n (T kq, c kk, c kk,..., c kk ), c qq, c qq,..., c qq )). = p n (p n (T kq, c kk, c kk,..., c kk ), c qq, c qq,..., c qq ). It follows that p n (p n (H T kq, c kk, c kk,..., c kk ), c qq, c qq,..., c qq ) = 0 and by (ii) of Definition 2.1 we obtain H kq = T kq. Finally let c tt R t and c kt M kt, k < t; by Lemma 3.1 we have ϕ(p n (H, c kt, c tt,..., c tt )) = ϕ(p n (T kk, c kt, c tt,..., c tt )) k 1 + ϕ(p n (T tt, c kt, c tt,..., c tt )) + ϕ(p n (T ik, c kt, c tt,..., c tt )) + n j=t+1 ϕ(p n (T tj, c kt, c tt,..., c tt )) = ϕ(p n (T kk, c kt, c tt,..., c tt )) k 1 + ϕ(p n (T tt, c kt, c tt,..., c tt )) + ϕ(p n (T ik, c kt, c tt,..., c tt )) + ϕ(0) = ϕ(p n (T kk, c kt, c tt,..., c tt )) + ϕ(p n (T tt, c kt, c tt,..., c tt )) ϕ(p n (T ik, c kt, c tt,..., c tt )). k 1 + Now let c kk R k ; by Lemma 3.1 and Lemma 4.4 we obtain ϕ(p n (p n (H, c kt, c tt,..., c tt ), c kk, c kk,..., c kk )) = ϕ(p n (p n (T kk, c kt, c tt,..., c tt ), c kk, c kk,..., c kk )) + ϕ(p n (p n (T tt, c kt, c tt,..., c tt ), c kk, c kk,..., c kk )) ϕ(p n (p n (T ik, c kt, c tt,..., c tt ), c kk, c kk,..., c kk )) k 1 + Since ϕ is injective we have = ϕ(p n (p n (T kk, c kt, c tt,..., c tt ), c kk, c kk,..., c kk ) k 1 + p n (p n (T tt, c kt, c tt,..., c tt ), c kk, c kk,..., c kk )) + ϕ(0) = ϕ(p n (p n (T kk + T tt, c kt, c tt,..., c tt ), c kk, c kk,..., c kk )). p n (p n (H (T kk + T tt ), c kt, c tt,..., c tt ), c kk, c kk,..., c kk ) = 0.

18 BRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR. By (ii) of Definition 2.1 it follows that (H kk T kk )c kt = c kt (H tt T tt ) for all n n c kt M kt. Therefore H ii = T ii + Z, where Z Z(T). Now by Lemma 4.2 the result is true. We are ready to prove our Theorem 4.1. Proof of Theorem 4.1. Let A, B T. By the previous lemmas we have ϕ(a + B) = ϕ( A ij + B ij ) = ϕ( = = n (A kk + B kk ) + n ϕ((a kk + B kk )) + n ϕ(a kk ) + + = ϕ( n A kk + + ϕ( n B kk + (A ij + B ij )) ϕ((a ij + B ij )) + S A,B n ϕ(b kk ) + n S Rk + ϕ(b ij ) + S A,B ϕ(a ij )) S A ϕ(b ij )) S B + ϕ(a ij ) n S Rk + S A,B = ϕ(a) + ϕ(b) + S A,B, n where S A,B = S Rk S A S B + S A,B, so the theorem is proved. 5. Final remarks Corollary 5.1. Let T be the triangular n-matrix unital ring and S be a ring satisfying the condition If p 2 (s, S) Z(S) then s Z(S). If ϕ : T S is a bijective Lie n-multiplicative mapping then ϕ is almost additive. Proof. Indeed, let ϕ : T S be a bijective Lie n-multiplicative mapping, T 2,..., T n T, and Z Z(T); we have p n (ϕ(z), ϕ(t 2 ),..., ϕ(t n )) = ϕ(p n (Z, T 2,..., T n )) = ϕ(0) = 0. Since T 2,... T n are arbitrary and ϕ is surjective it follows that ϕ(z) Z(S).

LIE n-multiplicative MAPPINGS ON TRIANGULAR n-matrix RINGS 19 Proposition 5.1. For any prime ring R, the statement If p 2 (r, R) Z(R) then r Z(R) holds true. Proof. See [5, Lemma 3]. Theorem 5.1. Let T be a triangular n-matrix unital ring and S be a prime ring. Then any bijective Lie n-multiplicative mapping from T onto S is almost additive. Corollary 5.2 (Xiaofei Qi and Jinchuan Hou [6]). Let A and B be unital algebras over a commutative ring R, and let M be a (A, B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri(A, M, B) be the triangular algebra and V any algebra over R. Assume that Φ : U V is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST T S) = Φ(S)Φ(T ) Φ(T )Φ(S) S, T U. Then Φ(S + T ) = Φ(S) + Φ(T ) + Z S,T for all S, T U, where Z S,T is an element in the centre Z(V) of V depending on S and T. Proof. This is consequence of our Theorem 4.1 for n = 2. 6. Application in nest algebras A nest N is a totally ordered set of closed subspaces of a Hilbert space H such that {0}, H N, and N is closed under the taking of arbitrary intersections and closed linear spans of its elements. The nest algebra associated to N is the set T (N ) = {T B(H) : T N N for all N N }, where B(H) is the algebra of bounded operators over a complex Hilbert space H. We recall the standard result ( ([1, Proposition ) 16]) that says that we can view A M T (N ) as a triangular algebra where A, B are themselves nest algebras. B Proposition 6.1. If N N \ {0, H} and E is the orthonormal projection onto N, then EN E and (1 E)N (1 E) are nest, T (EN E) = ET (N )E, and T ((1 E)N (1 E)) = (1 E)T (N )(1 E). Furthermore, ( ) T (EN E) ET (N )(1 E) T (N ) =. T ((1 E)N (1 E)) We refer the reader to [2] for the general theory of nest algebras. Corollary 6.1. Let P n be an increasing sequence of finite dimensional subspaces such that their union is dense in H. Consider P = {{0}, P n, n 1, H} a nest and T (P) the set consists of all operators which have a block upper triangular matrix with respect to P. If a mapping ϕ : T (P) T (P) satisfies ϕ(p 2 ([f, g]) = p 2 (ϕ(f), ϕ(g)) for all f, g T (P), then ϕ is almost additive.

20 BRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR. References [1] W.-S. Cheung, Commuting maps of triangular algebras, J. London Math. Soc. 63 (2001), 117 127. MR 1802761. [2] K. R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, 191, Longman Scientific & Technical, 1988. MR 0972978. [3] B. L. M. Ferreira, Multiplicative maps on triangular n-matrix rings, Int. J. Math. Game Theory Algebra 23 (2014), 1 14. [4] W. S. Martindale III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21 (1969), 695 698. MR 0240129. [5] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093 1100. MR 0095863. [6] X. Qi and J. Hou, Additivity of Lie multiplicative maps on triangular algebras, Linear Multilinear Algebra 59 (2011), 391 397. MR 2802521. B. L. M. Ferreira Universidade Tecnológica Federal do Paraná, Avenida Professora Laura Pacheco Bastos, 800, 85053-510, Guarapuava, Brazil brunoferreira@utfpr.edu.br H. Guzzo Jr. Universidade de São Paulo, Instituto de Matemática e Estatística, Rua do Matão, 1010, 05508-090, São Paulo, Brazil guzzo@ime.usp.br Received: January 18, 2018 Accepted: July 17, 2018