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Let R be a commutative ring with identity, and A a unital algebra over R. For any X,Y∈A, denote the Jordan product of X,Y by X∘Y=XY+YX. An additive mapping Δ from A into itself is called a derivation (resp., anti-derivation) if Δ(XY)=Δ(X)Y+XΔ(Y) (resp., Δ(XY)=Δ(Y)X+YΔ(X)) for all X,Y∈A. It is called a Jordan derivation if Δ(X∘Y)=Δ(X)∘Y+X∘Δ(Y) for all X,Y∈A. It is called a Jordan triple derivation if Δ(X∘Y∘Z)=Δ(X)∘Y∘Z+X∘Δ(Y)∘Z+X∘Y∘Δ(Z) for all X,Y,Z∈A. Obviously, every derivation or anti-derivation is a Jordan derivation. However, the inverse statement is not true in general (see [1]). If a Jordan derivation or Jordan triple derivation is not a derivation, then it is said to be proper. Otherwise, it is said to be improper.
In the past few decades, the problem of characterizing the structure of Jordan derivations and Jordan triple derivations has attracted the attention of many mathematical workers and has achieved some important research results. For example, Herstein in [2] proved that every Jordan derivation on a prime ring not of characteristic 2 is a derivation. This result was extended by Cusack in [3] and Brešar in [4] to the case of semiprime. Zhang in [5,6] showed that every Jordan derivation on a nest algebra or a 2-torsion free triangular algebra is a inner derivation or a derivation, respectively. Later, Hoger in [7] extended the result of Zhang in [6] and proved that, under certain conditions, each Jordan derivation on trivial extension algebras is a sum of a derivation and an anti-derivation. In addition, there have been many research results on Jordan triple derivations, as shown in references [8,9,10,11].
Definition 1.1. Let R be a commutative ring with identity, A a unital algebra over R, N0 be the set of all nonnegative integers, and D={dn}n∈N0 be a family of additive maps on A such that d0=idA (the identity map on A). D is said to be:
(i) a higher derivation if for each n∈N0,
dn(XY)=∑p+q=ndp(X)dq(Y) |
for all X,Y∈A;
(ii) a higher anti-derivation if for each n∈N0,
dn(XY)=∑p+q=ndp(Y)dq(X) |
for all X,Y∈A;
(iii) a higher Jordan derivation if for each n∈N0,
dn(X∘Y)=∑p+q=ndp(X)∘dq(Y) |
for all X,Y∈A;
(iv) a higher Jordan triple derivation if for each n∈N0,
dn(X∘Y∘Z)=∑p+q+r=ndp(X)∘dq(Y)∘dr(Z) |
for all X,Y,Z∈A.
If a higher Jordan derivation or a higher Jordan triple derivation is not a higher derivation, then it is said to be proper. Otherwise, it is said to be improper. With the deepening of research on this topic, many research achievements have been obtained about higher Jordan derivations and higher Jordan triple derivations. For example, Xiao and Wei in [12] proved that every higher Jordan derivation on triangular algebras is a higher derivation; Fu, Xiao, and Du in [13] extended this conclusion, and proved that every nonlinear higher Jordan derivation on triangular algebras is a higher derivation. Later, Vishki, Mirzavaziri, and Moafian in [14] proved that, under certain conditions, every higher Jordan derivation on trivial extension algebras is a higher derivation, and this conclusion further extended the works of the authors of references [12,13]. Salih and Haetinger in [15] proved that, under certain conditions, every higher Jordan triple derivation on prime rings is a higher derivation. Ashraf and Jabeenin [16] proved that every nonlinear higher Jordan triple derivable mapping on triangular algebras is a higher derivation.
In this paper, we are interested in describing the form of higher Jordan triple derivation on trivial extension algebras. As a main result, we give conditions under which each higher Jordan triple derivation on trivial extension algebras is a sum of a higher derivation and a higher anti-derivation. This result extends the study of Jordan derivation on trivial extension algebras [7], Jordan triple derivations on ∗-type trivial extension algebras [17], and Jordan higher derivations on trivial extension algebras [14].
Let R be a commutative ring with identity, A a unital algebra over R and M be an A-bimodule. Then the direct product A⊕M together with the pairwise addition, scalar product, and algebra multiplication defined by
(a,m)(b,n)=(ab,an+mb)(∀a,b∈A,m,n∈M) |
is an R-algebra with a unity (1,0) denoted by
T=A⊕M={(a,m):a∈A,m∈M} |
and T is called a trivial extension algebra.
An important example of trivial extension algebra is the triangular algebra which was introduced by Cheung in [18]. Let A and B be unital algebras over a commutative ring R, and M be a unital (A,B)-bimodule, which is faithful as both a left A-module and a right B-module. Then, the R-algebra
U=Tri(A,M,B)={(am0b):a∈A,m∈M,b∈B} |
under the usual matrix operations is called a triangular algebra. Basic examples of triangular algebras are upper triangular matrix algebras and nest algebras.
It is well-known that every triangular algebra can be viewed as a trivial extension algebra. Indeed, denote by A⊕B the direct product as an R-algebra, and then M is viewed as an A⊕B-bimodule with the module action given by (a,b)m=am and m(a,b)=mb for all (a,b)∈A⊕B and m∈M. Then triangular algebra U is isomorphic to trivial extensions algebra T=(A⊕B)⊕M. However, a trivial extension algebra is not necessarily a triangular algebra. For more details about trivial extension algebras, we refer the readers to [19,20,21].
The following notations will be used in our paper: Let R be a commutative ring with identity, A a unital algebra over R, M an A-bimodule, T=A⊕M be a 2-torsion free trivial extension algebra (i.e., for any X∈T, 2X={0} implies X=0), and denote by 1 and 0 are the unity and zero of T=A⊕M, respectively.
We say T=A⊕M is a ∗-type trivial extension algebra if A has a non-trivial idempotent element e and f=1−e such that
(i) eMf=M;
(ii) exeM={0} implies exe=0,∀x∈A;
(iii) Mfxf={0} implies fxf=0,∀x∈A;
(iv) exfye=0=fxeyf=0,∀x,y∈A.
For convenience, in the following we let P1=(e,0), P2=(f,0), and
Tij=PiTPj( 1≤i≤j≤2). |
It is not hard to see that the trivial extension algebra T may be represented as
T=P1TP1+P1TP2+P2TP1+P2TP2=T11+T12+T21+T22. |
Then every element A∈T may be represented as A=A11+A12+A21+A22, where Aij∈Tij(1≤i≤j≤2). In the following, we give a property of ∗-type trivial extension algebras (see Lemma 1.1).
Lemma 1.1. [17] Let T be a ∗-type trivial extension algebra and 1≤i≠j≤2. Then,
(i) for any A11∈T11, if A11T12=0, then A11=0;
(ii) for any A22∈T22, if T12A22=0, then A22=0;
(iii) AijBji=AiiBji=AijBii=0, ∀Aii,Bii∈Tii,∀Aij∈Tij,∀Bji∈Tji.
For ease of reading, we provide the main conclusions of reference [17] as follows:
Theorem 1.1. [17] Let T=A⊕M be a 2-torsion free ∗-type trivial extension algebra and Δ be a Jordan triple derivation on T. Then, there exists a derivation D and an anti-derivation φ on T, respectively, such that
Δ(A)=D(A)+φ(A) |
for all A∈T.
The main result of this paper is the following theorem:
Theorem 2.1. Let T=A⊕M be a 2-torsion free ∗-type trivial extension algebra, and D={dn}n∈N0 be a higher Jordan triple derivation on T. Then, there exists a higher derivation G={gn}n∈N0 and a higher anti-derivation F={fn}n∈N0 on T, respectively, such that
dn(X)=gn(X)+fn(X) |
for any n≥1 and X∈T.
In order to prove Theorem 2.1, we shall establish Theorems 2.2 and 2.3 in the following. We assume that T is a ∗-type trivial extension algebra, N0 is the set of all nonnegative integers, and D={dn}n∈N0 is a higher Jordan triple derivation on T.
In [17], it is proved that if d1 is a Jordan triple derivation on T, then for all Aij∈Tij (1≤i,j≤2), d1 satisfies the following properties (L):
(i) d1(P1)=−d1(P2);
(ii) d1(P1)=P1d1(P1)P2+P2d1(P1)P1 and d1(P2)=P1d1(P2)P2+P2d1(P2)P1;
(iii) P2d1(A11)P2=0, P1d1(A11)P2=A11d1(P1) and P2d1(A11)P1=d1(P1)A11;
(iv) P1d1(A22)P1=0,P1d1(A22)P2=d1(P2)A22 and P2d1(A22)P1=A22d1(P2);
(v) d1(A12)=P1d1(A12)P2+P2d1(A12)P1 and d1(A21)=P1d1(A21)P2+P2d1(A21)P1;
(vi) d1(P1)∘d1(P2)=d1(P1)∘d1(A12)=d1(P1)∘d1(A21)=d1(P2)∘d1(A12)=d1(P2)∘d1(A21)=0;
(vii) d1(A12)∘d1(A12)=d1(A21)∘d1(A21)=d1(A12)∘d1(A21)=0.
Now, for all Aij∈Tij (1≤i,j≤2), we assume that dk (1⩽k<n) satisfy the properties L. In the following, we show that dn satisfies the properties L.
Lemma 2.1. Let D={dn}n∈N0 be a higher Jordan triple derivation on T. Then, for each n≥1, and for any A11∈T11,A12∈T12,A21∈T21, A22∈T22,
(i) dn(P1)=P1dn(P1)P2+P2dn(P1)P1 and dn(P2)=P1dn(P2)P2+P2dn(P2)P1;
(ii) dn(P1)=−dn(P2);
(iii) P2dn(A11)P2=0, P1dn(A11)P2=A11dn(P1) and P2dn(A11)P1=dn(P1)A11;
(iv) P1dn(A22)P1=0,P1dn(A22)P2=dn(P2)A22 and P2dn(A22)P1=A22dn(P2);
(v) dn(A12)=P1dn(A12)P2+P2dn(A12)P1 and dn(A21)=P1dn(A21)P2+P2dn(A21)P1;
(vi) dn(P1)∘dn(P2)=dn(P1)∘dn(A12)=dn(P1)∘dn(A21)=dn(P2)∘dn(A12)=dn(P2)∘dn(A21)=0;
(vii) dn(A12)∘dn(A12)=dn(A21)∘dn(A21)=dn(A12)∘dn(A21)=0.
Proof. (i) For each n≥1 and for any X,Y,Z∈T, by the definition of D={dn}n∈N0, we get
dn(X∘Y∘Z)=∑p+q+r=ndp(X)∘dq(Y)∘dr(Z). | (2.1) |
Taking X=Y=Z=P1 in Eq (2.1), we assume that dk (1⩽k<n) satisfy the properties L, and then it follows from Lemma 1.1 (iii) that
4dn(P1)=∑p+q+r=ndp(P1)∘dq(P1)∘dr(P1)=∑p+q+r=n,1≤p,q,rdp(P1)∘dq(P1)∘dr(P1)+∑q+r=n,1≤q,rP1∘dq(P1)∘dr(P1)+∑p+r=n,1≤p,rdp(P1)∘P1∘dr(P1)+∑p+q=n,1≤p,qdp(P1)∘dq(P1)∘P1+dn(P1)∘P1∘P1+P1∘dn(P1)∘P1+P1∘P1∘dn(P1)=dn(P1)∘P1∘P1+P1∘dn(P1)∘P1+P1∘P1∘dn(P1)=4P1dn(P1)P1+4P1dn(P1)+4dn(P1)P1. |
This yields from the 2-torsion freeness of T that
P1dn(P1)P1=P2dn(P1)P2=0. |
Similarly, we get that
P1dn(P2)P1=P2dn(P2)P2=0. |
Therefore, dn(P1)=P1dn(P1)P2+P2dn(P1)P1 and dn(P2)=P1dn(P2)P2+P2dn(P2)P1.
(ii) For each n≥1, taking X=P1,Y=P2,Z=P1 in Eq (2.1), we assume that dk (1⩽k<n) satisfy the properties L, then by Lemma 1.1 (iii) and Lemma 2.1 (i), we get that
0=∑p+q+r=ndp(P1)∘dq(P2)∘dr(P1)=∑p+q+r=n,1≤p,q,rdp(P1)∘dq(P2)∘dr(P1)+∑q+r=n,1≤q,rP1∘dq(P2)∘dr(P1)+∑p+r=n,1≤p,rdp(P1)∘P2∘dr(P1)+∑p+q=n,1≤p,qdp(P1)∘dq(P2)∘P1+dn(P1)∘P2∘P1+P1∘dn(P2)∘P1+P1∘P2∘dn(P2)=dn(P1)∘P2∘P1+P1∘dn(P2)∘P1+P1∘P2∘dn(P2)={dn(P1)P2+P2dn(P1)}∘P1+{P1dn(P2)+dn(P2)P1}∘P1=P1dn(P1)P2+P2dn(P1)P1+P1dn(P2)+dn(P2)P1+2P1dn(P2)P1=P1dn(P1)P2+P2dn(P1)P1+P1dn(P2)P2+P2dn(P2)P1=dn(P1)+dn(P2). |
(iii)–(iv) For each n≥1 and for any A11∈T11, taking X=A11,Y=Z=P2 in Eq (2.1), we assume that dk (1⩽k<n) satisfy the properties L, and then by Lemma 1.1 (iii) and Lemma 2.1 (i,ii), we get that
0=∑p+q+r=ndp(A11)∘dq(P2)∘dr(P2)=∑p+q+r=n,1≤p,q,rdp(A11)∘dq(P2)∘dr(P2)+∑q+r=n,1≤q,rA11∘dq(P2)∘dr(P2)+∑p+r=n,1≤p,rdp(A11)∘P2∘dr(P2)+∑p+q=n,1≤p,qdp(A11)∘dq(P2)∘P2+dn(A11)∘P2∘P2+A11∘dn(P2)∘P2+A11∘P2∘dn(P2)=dn(A11)∘P2∘P2+A11∘dn(P2)∘P2+A11∘P2∘dn(P2)={dn(A11)P2+P2dn(A11)}∘P2+{A11dn(P2)+dn(P2)A11}∘P2={dn(A11)P2+P2dn(A11)+2P2dn(A11)P2}+{A11dn(P2)+P2dn(P2)A11}=dn(A11)P2+P2dn(A11)+A11dn(P2)+dn(P2)A11. |
This implies that P2dn(A11)P2=0. and
P1dn(A11)P2=A11dn(P1) and P2dn(A11)P1=dn(P1)A11. |
Similarly, for each n≥1 and for any A22∈T22, we get that P1dn(A22)P1=0, P1dn(A22)P2=dn(P2)A22 and P2dn(A22)P1=A22dn(P2).
(v) For each n≥1 and for any A12∈T12, taking X=P1,Y=A12,Z=P2 in Eq (2.1), we assume that dk (1⩽k<n) satisfy the properties L, and then by Lemma 1.1 (iii) and Lemma 2.1 (i,ii), we get that
dn(A12)=∑p+q+r=ndp(P1)∘dq(A12)∘dr(P2)=∑p+q+r=n,1≤p,q,rdp(P1)∘dq(A12)∘dr(P2)+∑q+r=n,1≤q,rP1∘dq(A12)∘dr(P2)+∑p+r=n,1≤p,rdp(P1)∘A12∘dr(P2)+∑p+q=n,1≤p,qdp(P1)∘dq(A12)∘P2+dn(P1)∘A12∘P2+P1∘dn(A12)∘P2+P1∘A12∘dn(P2)=P1∘dn(A12)∘P2=P1dn(A12)P2+P2dn(A12)P1. |
Similarly, for each n≥1 and for any A21∈T21, we get that dn(A21)=P1dn(A21)P2+P2dn(A21)P1.
(vi) For each n≥1 and for any A12∈T12, A21∈T21, by Lemma 1.1 (iii) and Lemma 2.1 (i,ii,v), we can easily check that (vi) holds. Similarly, we show (vii) holds. The proof is complete.
Theorem 2.2. Let F={fn}n∈N0 be a sequence of mappings on T (with f0=ifT). For each n≥1 and X∈T, define
fn(X)=P2dn(P1XP2)P1+P1dn(P2XP1)P2. |
Then, F is a higher anti-derivation on T.
It is clear that fn(Aii)=0 and fn(Aij)=Pjfn(Aij)Pi for each n≥1, and for any Aii∈Tii,Aij∈Tij (1≤i≠j≤2).
In the following, we show that F={fn}n∈N0 is a higher anti-derivation, i.e., for each n≥1 and for any X,Y∈T, fn satisfies fn(XY)=∑p+q=nfp(Y)fq(X). For this, we introduce Lemmas 2.2 and 2.3, and prove Lemmas 2.2 and 2.3.
Lemma 2.2. Let fn:T→T be defined as in Theorem 2.2. Then, for each n≥1 and for any Aii,Bii∈Tii,Aij,Bij∈Tij,Bji∈Tji,Bjj∈Tjj (1≤i≠j≤2),
(i) fn(AiiBii)=∑p+q=nfp(Bii)fq(Aii);
(ii) fn(AiiBjj)=∑p+q=nfp(Bjj)fq(Aii);
(iii) fn(AiiBji)=∑p+q=nfp(Bji)fq(Aii);
(iv) fn(AijBii)=∑p+q=nfp(Bii)fq(Aij);
(v) fn(AijBij)=∑p+q=nfp(Bij)fq(Aij);
(vi) fn(AijBji)=∑p+q=nfp(Bji)fq(Aij).
Proof. (i) For any n≥1 and Aii,Bii∈Tii (1≤i≤2), we get from fn(AiiBii)=fn(Aii)=fn(Bii)=0 that
fn(AiiBii)=∑p+q=nfp(Bii)fq(Aii). |
Similarly, we can show (ii) holds.
(iii) For each n≥1 and for any Aii∈Tii,Bji∈Tji (1≤i≠j≤2), on the one hand, we have fn(AiiBji)=fn(0)=0. On the other hand, it follows from fn(Aii)=0 and fn(Bji)=Pifn(Bji)Pj that
∑p+q=nfp(Bji)fq(Aii)=∑p+q=n,1≤p,qfp(Bji)fq(Aii)+fn(Bji)Aii+Bjifn(Aii)=fn(Bji)Aii=(Pifn(Bji)Pj)Aii=0. |
Therefore, fn(AiiBji)=∑p+q=nfp(Bji)fq(Aii). Similarly, we get (iv).
(v) For each n≥1 and for any Aij,Bij∈Tij (1≤i≠j≤2), on the one hand, we have fn(AijBij)=fn(0)=0. On the other hand, we get from fn(Bij)fn(Aij)={Pjfn(Bij)Pi}{Pjfn(Aij)Pi}=0 and Lemma 1.1 (iii) that
∑p+q=nfp(Bij)fq(Aij)=∑p+q=n,1≤p,qfp(Bij)fq(Aij)+fn(Bij)Aij+Bijfn(Aij)=fn(Bij)Aij+Bijfn(Aij)={Pjfn(Bij)Pi}Aij+Bij{Pjfn(Aij)Pi}=0. |
Therefore, fn(AijBij)=∑p+q=nfp(Bij)fq(Aij). Similarly, we get (vi). The proof is complete.
Lemma 2.3. Let fn:T→T be defined as in Theorem 2.2. Then, for each n≥1 and for any Aii∈Tii,Bij∈Tij,Bjj∈Tjj (1≤i≠j≤2),
(i) fn(AiiBij)=∑p+q=nfp(Bij)fq(Aii);
(ii) fn(AijBjj)=∑p+q=nfp(Bjj)fq(Aij).
Proof. (i) For each n≥1 and for any Aii∈Tii,Bij∈Tij (1≤i≠j≤2), it follows from AiiBij=Aii∘Bij∘Pj and Lemma 2.1 that
fn(AiiBij)=Pjdn(AiiBij)Pi=Pjdn(Aii∘Bij∘Pj)Pi=Pj{∑p+q+r=ndp(Aii)∘dq(Bij)∘dr(Pj)}Pi=Pj{∑p+q+r=n,1≤p,q,rdp(Aii)∘dq(Bij)∘dr(Pj)}Pi+Pj{∑q+r=n,1≤q,rAii∘dq(Bij)∘dr(Pj)}Pi+Pj{∑p+r=n,1≤r,tdp(Aii)∘Bij∘dr(Pj)}Pi+Pj{∑p+q=n,1≤r,s,tdp(Aii)∘dq(Bij)∘Pj}Pi+Pj{dn(Aii)∘Bij∘Pj}Pi+Pj{Aii∘dn(Bij)∘Pj}Pi+Pj{Aii∘Bij∘dn(Pj)}Pi=Pj{dn(Aii)∘Bij∘Pj}Pi+Pj{Aii∘dn(Bij)∘Pj}Pi+Pj{Aii∘Bij∘dn(Pj)}Pi=Pj{dn(Aii)BijPj+Pjdn(Aii)Bij+Bijdn(Aii)Pj}Pi+Pj{Aiidn(Bij)Pj+Pjdn(Bij)Aii}Pi+Pj{AiiBij∘dn(Pj)}Pi=Pjdn(Bij)AiiPi=fn(Bij)Aii. | (2.2) |
On the other hand, it is follows from fn(Aii)=0 (n≥1) that fn−k(Bij)dk(Aii)=0, so we get
fn(AiiBij)=fn(Bij)Aii+fn−1(Bij)f1(Aii)+fn−2(Bij)f2(Aii)+…+Bijfn(Aii)=∑p+q=nfp(Bij)fq(Aii). |
Similarly, we get (ii). The proof is complete.
In the following, we give the completed proof of Theorem 2.2:
Proof of Theorem 2.2. For each n≥1, let A=A11+A12+A21+A22 and B=B11+B12+B21+B22 be arbitrary elements of T, where Aij,Bij∈Tij (1≤i,j≤2). Then, it follows from Lemmas 2.2 and 2.3 that
fn(AB)=fn((A11+A12+A21+A22)(B11+B12+B21+B22))=fn(A11B11)+fn(A11B12)+fn(A11B21)+fn(A11B22)+fn(A12B11)+fn(A12B12)+fn(A12B21)+fn(A12B22)+fn(A21B11)+fn(A21B12)+fn(A21B21)+fn(A21B22)+fn(A22B11)+fn(A22B12)+fn(A22B21)+fn(A22B22)=∑p+q=nfp(B11)fq(A11)+∑p+q=nfp(B11)fq(A12)+∑p+q=nfp(B11)fq(A21)+∑p+q=nfp(B11)fq(A22)+∑p+q=nfp(B12)fq(A11)+∑p+q=nfp(B12)fq(A12)+∑p+q=nfp(B12)fq(A21)+∑p+q=nfp(B12)fq(A22)+∑p+q=nfp(B21)fq(A11)+∑p+q=nfp(B21)fq(A12)+∑p+q=nfp(B21)fq(A21)+∑p+q=nfp(B21)fq(A22)+∑p+q=nfp(B22)fq(A11)+∑p+q=nfp(B22)fq(A12)+∑p+q=nfp(B22)fq(A21)+∑p+q=nfp(B22)fq(A22)=∑p+q=nfp(B)fq(A). |
Therefore, F={fn}n∈N0 is a higher anti-derivation on T. The proof is complete.
Theorem 2.3. Let G={gn}n∈N0 be a sequence of mappings on T (with g0=igT). For each n≥1 and for any X∈T, define
gn(X)=dn(X)−fn(X). |
Then, G is a higher derivation on T.
Next, we show that G={gn}n∈N0 is a higher derivation on T. In order to prove G is a higher derivation, we introduce Lemmas 2.4–2.6, and then, using the mathematical induction, we prove Lemmas 2.4–2.6.
In [12] Theorem 1.3, we have proved that if g1=d1−f1, then g1 is a derivation on T, i.e., for any X,Y∈T, g1 satisfies
g1(XY)=g1(X)Y+Xg1(Y)=∑p+q=1gp(X)gq(Y). |
Therefore, in the following, we assume that
gk(XY)=∑p+q=kgp(X)gq(Y) | (2.3) |
for each 1⩽k<n and X,Y∈T. Next, we prove that Lemmas 2.4–2.6 hold.
By the definitions of F={fn}n∈N0 and G={gn}n∈N0, and by Lemma 2.1, we can easily check that the following Lemma holds:
Lemma 2.4. Let gn:T→T be defined as in Theorem 2.3. Then, for each n≥1 and for any Aii∈Tii,Aij∈Tij (1≤i≠j≤2),
(i) gn(Pi)=−gn(Pj) and gn(Pi)=Pign(Pi)Pj+Pjgn(Pi)Pi;
(ii) Pjgn(Aii)Pj=0,Pign(Aii)Pj=Aiign(Pi) and Pjgn(Aii)Pi=gn(Pi)Aii;
(iii) gn(Aij)=Pign(Aij)Pj.
Lemma 2.5. Let gn:T→T be defined as in Theorem 2.3. Then, for each n≥1, and for any Aii,Bii∈Tii, Bjj∈Tjj,Aij,Bij∈Tij (1≤i≠j≤2),
(i) gn(AiiBij)=∑p+q=ngp(Aii)gq(Bij);
(ii) gn(AijBjj)=∑p+q=ngp(Aij)gq(Bjj);
(iii) gn(AiiBii)=∑p+q=ngp(Aii)gq(Bii);
(iv) gn(AiiBjj)=∑p+q=ngp(Aii)gqBjj).
Proof. (i) For each n≥1 and for any Aii∈Tii,Bij∈Tij (1≤i≠j≤2), taking X=Aii,Y=Bij,Z=Pj in Eq (2.1), and by Lemma 1.1 (iii) and Lemma 2.1, we get
dn(AiiBij)=dn(Aii∘Bij∘Pj)=∑p+q+r=ndp(Aii)∘dq(Bij)∘dr(Pj)=∑p+q+r=n,1≤p,q,rdp(Aii)∘dq(Bij)∘dr(Pj)+∑q+r=n,1≤q,rAii∘dq(Bij)∘dr(Pj)+∑p+r=n,1≤p,rdp(Aii)∘Bij∘dr(Pj)+∑p+q=n,1≤p,qdp(Aii)∘dq(Bij)∘Pj+dn(Aii)∘Bij∘Pj+Aii∘dn(Bij)∘Pj+Aii∘Bij∘dn(Pj)=∑p+q=n,1≤p,qdp(Aii)∘dq(Bij)∘Pj+dn(Aii)∘Bij∘Pj+Aii∘dn(Bij)∘Pj=∑p+q=n,1≤p,qdp(Aii)∘dq(Bij)+dn(Aii)Bij+Aiidn(Bij)+dn(Bij)Aii=∑p+q=ndp(Aii)dq(Bij)+dn(Bij)Aii. |
Therefore, it follows from Eq (2.2), with fn(Aii)=0 and fn(Aij)=Pjfn(Aij)Pi (n≥1), that
gn(AiiBij)=dn(AiiBij)−fn(AiiBij)=∑p+q=ndp(Aii)dq(Bij)+dn(Bij)Aii−dn(Bij)Aii=∑p+q=n,1≤p,qdp(Aii)dq(Bij)+dn(Aii)Bij+Aiidn(Bij)=∑p+q=n,1≤p,q{dp(Aii)−fp(Aii)}dq(Bij)+{dn(Aii)−fn(Aii)}Bij+Aii{dn(Bij)−fn(Bij)}=∑p+q=n,1≤p,qgp(Aii)dq(Bij)+gn(Aii)Bij+Aiign(Bij)=∑p+q=n,1≤p,qgp(Aii){dq(Bij)−fq(Bij)}+gn(Aii)Bij+Aiign(Bij)=∑p+q=n,1≤p,qgp(Aii)gq(Bij)+gn(Aii)Bij+Aiign(Bij)=∑p+q=ngp(Aii)gq(Bij). |
Similarly, we get that (ii) holds.
(iii) For each n≥1 and for any Aii,Bii∈Tii,Xij∈Tij (1≤i≠j≤2), by Lemma 2.5 (i) and Eq (2.3), on the one hand, we get
gn(AiiBiiXij)=gn((AiiBii)Xij)=∑p+q=n,1≤qgp(AiiBii)gq(Xij)+gn(AiiBii)Xij=∑p+q=n,1≤q{∑r+s=pgr(Aii)gs(Bii)}gq(Xij)+gn(AiiBii)Xij=∑r+s+q=n,1≤qgr(Aii)gs(Bii)gq(Xij)+gn(AiiBii)Xij. |
On the other hand, we have
gn(AiiBiiXij)=gn(Aii(BiiXij))=∑p+q=ngp(Aii)gq(BiiXij)=∑p+q=ngp(Aii)∑r+s=qgr(Bii)gs(Xij)=∑p+r+s=ngp(Aii)gr(Bii)gs(Xij)=∑p+r+s=n,1≤sgp(Aii)gr(Bii)gs(Xij)+∑p+r=ngp(Aii)gr(Bii)Xij. |
Comparing the above two equations, we get
{gn(AiiBii)−∑p+r=ngp(Aii)gr(Bii)}Xij=0,∀Xij∈Tij(1≤i≠j≤2). |
This yields from Lemma 1.1 (i) that
Pign(AiiBii)Pi=Pi{∑p+r=ngp(Aii)gr(Bii)}Pi. | (2.4) |
Next, we show that
Pign(AiiBii)Pj=Pi{∑p+r=ngp(Aii)gr(Bii)}Pj and Pjgn(AiiBii)Pi=Pj{∑p+r=ngp(Aii)gr(Bii)}Pi. |
Indeed, for each n≥1 and for any Aii,Bii∈Tii (1≤i≠j≤2), taking X=Aii,Y=Z=Pj in Eq (2.1), by Lemma 2.1, we get
0=dn(Aii∘Pj∘Pj)=∑p+q+r=ndp(Aii)∘dq(Pj)∘dr(Pj)=∑p+q+r=n,1≤p,q,rdp(Aii)∘dq(Pj)∘dr(Pj)+∑q+r=n,1≤q,rAii∘dq(Pj)∘dr(Pj)+∑p+r=n,1≤p,rdp(Aii)∘Pj∘dr(Pj)+∑p+q=n,1≤p,qdp(Aii)∘dq(Pj)∘Pj+dn(Aii)∘Pj∘Pj+Aii∘dn(Pj)∘Pj+Aii∘Pj∘dn(Pj)=∑p+q=n,1≤p,qdp(Aii)∘dq(Pj)∘Pj+dn(Aii)∘Pj∘Pj+Aii∘dn(Pj)∘Pj=∑p+q=n,1≤p,q{dp(Aii)dq(Pj)+dq(Pj)dp(Aii)}∘Pj+dn(Aii)Pj+Pjdn(Aii)+Aiidn(Pj)+dn(Pj)Aii=∑p+q=ndp(Aii)dq(Pj)+∑p+q=ndq(Pj)dp(Aii). |
Therefore, we get from ∑p+q=ndp(Aii)dq(Pj)∈Tij and ∑p+q=ndq(Pj)dp(Aii)∈Tji that
∑p+q=ndp(Aii)dq(Pj)=0 and ∑p+q=ndq(Pj)dp(Aii)=0. |
So we get from fk(Aii)=0 and fk(Pj)=0 (k≥1) that
0=∑p+q=ndp(Aii)dq(Pj)=∑p+q=n,1≤p,qdp(Aii)dq(Pj)+dn(Aii)Pj+Aiidn(Pj)=∑p+q=n,1≤p,q(dp(Aii)−fp(Aii))(dq(Pj)−fq(Pj))+(dn(Aii)−fn(Aii))Pj+Aii(dn(Pj)−fn(Pj))=∑p+q=n,1≤p,qgp(Aii)gq(Pj)+gn(Aii)Pj+Aiign(Pj)=∑p+q=ngp(Aii)gq(Pj)=∑p+q=n,1≤qgp(Aii)gq(Pj)+gn(Aii)Pj=−∑p+q=n,1≤qgp(Aii)gq(Pi)+gn(Aii)Pj=−∑p+q=ngp(Aii)gq(Pi)+gn(Aii)Pi+gn(Aii)Pj=−∑p+q=ngp(Aii)gq(Pi)+gn(Aii). |
Therefore,
gn(Aii)=∑p+q=ngp(Aii)gq(Pi). | (2.5) |
For each n≥1 and for any Aii,Bii∈Tii, by Eq (2.5), we get
gn(AiiBii)=∑p+q=ngp(AiiBii)gq(Pi)=∑p+q=n,1≤qgp(AiiBii)gq(Pi)+gn(AiiBii)Pi=∑p+q=n,1≤q{∑r+s=pgr(Aii)gs(Bii)}gq(Pi)+gn(AiiBii)Pi=∑r+s+q=n,1≤qgr(Aii)gs(Bii)gq(Pi)+gn(AiiBii)Pi=n∑r=0gr(Aii){∑s+q=n−r,1≤qgs(Bii)gq(Pi)}+gn(AiiBii)Pi=n∑r=0gr(Aii){∑s+q=n−rgs(Bii)gq(Pi)−gn−r(Bii)Pi}+gn(AiiBii)Pi=n∑r=0gr(Aii){gn−r(Bii)−gn−r(Bii)Pi}+gn(AiiBii)Pi=n∑r=0gr(Aii)gn−r(Bii)−n∑r=0gr(Aii)gn−r(Bii)Pi+gn(AiiBii)Pi=∑p+q=ngp(Aii)gq(Bii)+∑p+q=ngp(Aii)gq(Bii)Pi+gn(AiiBii)Pi. |
This implies that
Pign(AiiBii)Pj=Pi{∑p+q=ngp(Aii)gq(Bii)}Pj. | (2.6) |
Similarly, we get
Pjgn(AiiBii)Pi=Pj{∑p+q=ngp(Aii)gq(Bii)}Pi. | (2.7) |
Therefore, by Eqs (2.4), (2.6), (2.7) and Lemma 2.4 (ii), we get that
gn(AiiBii)=Pi{∑p+q=ngp(Aii)gq(Bii)}Pi+Pi{∑p+q=ngp(Aii)gq(Bii)}Pj+Pj{∑p+q=ngp(Aii)gq(Bii)}Pi=∑p+q=ngp(Aii)gq(Bii). |
(iv) For each n≥1 and for any Aii∈Tii,Bjj∈Tjj (1≤i≠j≤2), taking X=Aii,Y=Bjj,Z=Pj (1≤i≠j≤2) in Eq (2.1), we get from Lemma 2.1 that
0=dn(Aii∘Bjj∘Pj)=∑p+q+r=ndp(Aii)∘dq(Bjj)∘dr(Pj)=∑p+q+r=n,1≤p,q,rdp(Aii)∘dq(Bjj)∘dr(Pj)+∑q+r=n,1≤q,rAii∘dq(Bjj)∘dr(Pj)+∑p+r=n,1≤p,rdp(Aii)∘Bjj∘dr(Pj)+∑p+q=n,1≤p,qdp(Aii)∘dq(Bjj)∘Pj+dn(Aii)∘Bjj∘Pj+Aii∘dn(Bjj)∘Pj+Aii∘Bjj∘dn(Pj)=∑p+q=n,1≤p,qdp(Aii)∘dq(Bjj)∘Pj+dn(Aii)∘Bjj∘Pj+Aii∘dn(Bjj)∘Pj=∑p+q=n,1≤p,qdp(Aii)dq(Bjj)∘Pj+∑p+q=n,1≤p,qdp(Bjj)dq(Aii)∘Pj+dn(Aii)Bjj+Bjjdn(Aii)+Aiidn(Bjj)+dn(Bjj)Aii=∑p+q=n,1≤p,qdp(Aii)dq(Bjj)+∑p+q=n,1≤p,qdp(Bjj)dq(Aii)+dn(Aii)Bjj+Bjjdn(Aii)+Aiidn(Bjj)+dn(Bjj)Aii=∑p+q=ndp(Aii)dq(Bjj)+∑p+q=ndp(Bjj)dq(Aii). |
Hence, we get from ∑p+q=ndp(Aii)dq(Bjj)∈Tij and ∑p+q=ndp(Bjj)dq(Aii)∈Tji (1≤i≠j≤2) that
∑p+q=ndp(Aii)dq(Bjj)=∑p+q=ndp(Bjj)dq(Aii)=0. |
Therefore, it follows from gk(Aii)=dk(Aii) and gk(Bjj)=dk(Bjj) (k≥1) that
gn(AiiBjj)=0=∑p+q=ndp(Aii)dq(Bjj)=∑p+q=ngp(Aii)gq(Bjj). |
The proof is complete.
Lemma 2.6. Let gn:T→T be defined as in Theorem 2.3. Then for each n≥1 and for any Aii∈Tii,Bjj∈Tjj,Aij,Bij∈Tij,Aji,Bji∈Tji (1≤i≠j≤2),
(i) gn(AijBji)=∑p+qgp(Aij)gq(Bji);
(ii) gn(AijBij)=∑p+q=ngp(Aij)gq(Bij);
(iii) gn(AiiBji)=∑p+q=ngp(Aii)gq(Bji);
(iv) gn(AjiBjj)=∑p+q=ngp(Aji)gq(Bjj).
Proof. (i) For each n≥1 and for any Aij∈Tij,Bji∈Tji (1≤i≠j≤2), it follows from Lemma 1 (iii) that AijBji=0, and therefore we get
gn(AijBji)=gn(0)=0. |
On the other hand, by Lemma 1 (iii) and Lemma 2.4 (iii), we have gp(Aij)gq(Bji)=0, therefore we get
gn(AijBji)=0=∑p+q=ngp(Aij)gq(Bji). |
Similarly, we get that (ii) holds.
(iii) For each n≥1 and for any Aii∈Tii,Bji∈Tji (1≤i≠j≤2), by Lemma 1 (iii) and Lemma 2.4 (ii,iii), we get gp(Aii)gq(Bji)=0, and therefore we get
gn(AiiBji)=0=∑p+q=ngp(Aii)gq(Bji). |
Similarly, we get (iv) holds. The proof is complete.
In the following, we complete the proof of Theorem 2.3.
Proof of Theorem 2.3. For any n≥1, let A=A11+A12+A21+A22 and B=B11+B12+B21+B22 be arbitrary elements of T, where Aij,Bij∈Tij (1≤i,j≤2). It follows from Lemmas 2.4–2.6 that
gn(AB)=gn((A11+A12+A21+A22)(B11+B12+B21+B22))=gn(A11B11)+gn(A11B12)+gn(A11B21)+gn(A11B22)+gn(A12B11)+gn(A12B12)+gn(A12B21)+gn(A12B22)+gn(A21B11)+gn(A21B12)+gn(A21B21)+gn(A21B22)+gn(A22B11)+gn(A22B12)+gn(A22B21)+gn(A22B22)=∑p+q=ngp(A11)gq(B11)+∑p+q=ngp(A11)gq(B12)+∑p+q=ngp(A11)gq(B21)+∑p+q=ngp(A11)gq(B22)+∑p+q=ngp(A12)gq(B11)+∑p+q=ngp(A12)gq(B12)+∑p+q=ngp(A12)gq(B21)+∑p+q=ngp(A12)gq(B22)+∑p+q=ngp(A21)gq(B11)+∑p+q=ngp(A21)gq(B12)+∑p+q=ngp(A21)gq(B21)+∑p+q=ngp(A21)gq(B22)+∑p+q=ngp(A22)gq(B11)+∑p+q=ngp(A22)gq(B21)+∑p+q=ngp(A22)gq(B21)+∑p+q=ngp(A22)gq(B22)=∑p+q=ngp(A11+A12+A21+A22)gq(B11+B12+B21+B22)=∑p+q=ngp(A)gq(B). |
Therefore, G={gn}n∈N0 is a higher derivation on T. The proof is complete.
Next, we show that Theorem 2.1 holds.
Proof of Theorem 2.1. For each n≥1 and for any A,B∈T, by Theorems 2.2 and 2.3, we obtain that
dn(A)=gn(A)+fn(A), |
where G={gn}n∈N0 is a higher derivation and F={fn}n∈N0 is a higher anti-derivation from T into itself such that fn(Aii)=0 for all Aii∈Tii (1≤i≤2). The proof is complete.
Remark 2.1. Let D={dn}n∈N0 be a higher Jordan triple derivation from T into itself. Then, by Theorems 2.1 and 2.2, we obtain that the following statements are equivalent.
(i) D={dn}n∈N0 is a higher derivation;
(ii) Pjdn(Aij)Pi=0 for each n≥1 and for any Aij∈Tij (1≤i≠j≤2);
(iii) dn(Aij)∈Tij for each n≥1 and for any Aij∈Tij (1≤i≠j≤2).
In the following, we show that every higher Jordan triple derivation on triangular algebras is a higher derivation.
Corollary 2.1. Let A and B be unital algebras over a commutative ring R and M be a unital (A,B)-bimodule, which is faithful as both a left A-module and a right B-module, and U be the 2-torsion free triangular algebra, and D={dn}n∈N0 be a higher Jordan triple derivation on U. Then D={dn}n∈N0 is a higher derivation.
Proof of Corollary 2.1. Let 1A and 1B be the identities of the algebras A and B, respectively, and let 1 be the identity of the triangular algebra U. We denote
P1=(1A000) by the standard idempotent of U, P2=1−P1=(0001B) |
and
Uij=PiUPj for 1≤i≤j≤2. |
It is clear that the triangular algebra U may be represented as
U=P1UP1+P1UP2+P2UP2=A+M+B. |
Here P1UP1 and P2UP2 are subalgebras of U isomorphic to A and B, respectively, and P1UP2 is a (P1UP1,P2UP2)-bimodule isomorphic to the (A,B)-bimodule M.
By the definition of triangular algebra U, we can easily check that U is a ∗-type trivial extension algebra, and so if U is a 2-torsion free triangular algebra, then for any n≥1, A=A11+A12+A22∈U, where Aij∈Uij (1≤i,j≤2), we get from Theorem 2.1 that
dn(A)=gn(A)+fn(A). |
Where G={gn}n∈N0 is a higher derivation and F={fn}n∈N0 is a higher anti-derivation from U into itself such that fn(Aii)=0 for all Aii∈Uii (1≤i≤2). Next, we show that fn(A12)=0 for each n≥1 and for any A12∈U12.
Indeed, for any A12∈U12, it follows from Lemma 2.1 (v) and U21={0} that
dn(A12)=P1dn(A12)P2+P2dn(A12)P1=P1dn(A12)P2. |
And then we obtain from the definition of fn in Theorem 2.2 that fn(A12)=P2dn(A12)P1=0. Therefore, for any A∈U, fn(A)=0, so D={gn}n∈N0 is a higher derivation. The proof is complete.
Next, we give an application of Corollary 2.1 to certain special classes of triangular algebras, such as block upper triangular matrix algebras and nest algebras.
Let R be a commutative ring with identity and let Mn×k(R) be the set of all n×k matrices over R. For n≥2 and m≤n, the block upper triangular matrix algebra Tˉkn(R) is a subalgebra of Mn(R) with the form
(Mk1(R)Mk1×k2(R)⋯Mk1×km(R)0Mk2(R)⋯Mk2×km(R)⋮⋮⋱⋮00⋯Mkm(R)), |
where ˉk=(k1,k2,⋯,km) is an ordered m-vector of positive integers such that k1+k2+⋯+km=n.
A nest of a complex Hilbert space H is a chain N of closed subspaces of H containing {0} and H, which is closed under arbitrary intersections and closed linear span, and B(H) is the algebra of all bounded linear operators on H. The nest algebra associated with N is the algebra
AlgN={T∈B(H):TN⊆N, for all N∈N}. |
A nest N is called trivial if N={0,H}. It is clear that every nontrivial nest algebra is a triangular algebra and every finite dimensional nest algebra is isomorphic to a complex block upper triangular matrix algebra.
Corollary 2.2. Let Tˉkn(R) be a 2-torsion free block upper triangular matrix algebra, and D={dn}n∈N0 be a higher Jordan triple derivation on Tˉkn(R). Then, D={dn}n∈N0 is a higher derivation.
Corollary 2.3. Let N be a nontrivial nest of a complex Hilbert space H, AlgN a nest algebra, and D={dn}n∈N0 a higher Jordan triple derivation on AlgN. Then, D={dn}n∈N0 is a higher derivation.
The authors declare that they have not used artificial intelligence tools in the creation of this article.
This research is supported by the National Natural Science Foundation of China (No.11901451), Talent Project Foundation of Yunnan Provincial Science and Technology Department (No.202105AC160089), Natural Science Foundation of Yunnan Province (No.202101BA070001198), and Basic Research Foundation of Yunnan Education Department (No.2021J0915).
The authors declare that there are no conflicts of interest regarding the publication of this paper.
[1] | Dejerine J (1914) Semiologie des affections du systeme nerveux. Paris: Masson. |
[2] | Goldstein K (1948) Language and language disturbances. New York: Grune & Stratton |
[3] | Head H (1926) Aphasia and kindred disorders of speech. London: Cambridge University Press. |
[4] | Pick A (1931) Aphasia. Springfiedl, Ill: Charles C. Thomas. |
[5] | Luria AR (1962) Higher Cortical Functions in Man. Moscow University Press. |
[6] | Luria AR (1970) Traumatic Aphasia: Its Syndromes, Psychology, and Treatment. Mouton de Gruyter. |
[7] | Luria AR (1973) The Working Brain. Basic Books. |
[8] | Luria AR (1976) Basic Problems in Neurolinguistics. New York: Mouton. |
[9] | Penfield W, Rasmussen T (1952) The cerebral cortex of man. MacMillan Company. |
[10] | Penfield W, Jaspers HH (1954) Epilepsy and the functional anatomy of the human brain. Boston: Little Brown |
[11] |
Ojemann GA (1983) Brain organization for language from the perspective of electrical stimulation mapping. Behav Brain Sci 6: 189-206. doi: 10.1017/S0140525X00015491
![]() |
[12] |
Price CJ (2010) The anatomy of language: a review of 100 fMRI studies published in 2009. Ann N Y Acad Sci 1191: 62-88. doi: 10.1111/j.1749-6632.2010.05444.x
![]() |
[13] | Gernsbacher MA, Kaschak MP (2003) Neuroimaging studies of language production and comprehension. An Rev Psychol 54: 91 |
[14] | Ardila A (2014) Aphasia Handbook. Miami, FL: Florida International University |
[15] | Benson DF (1979) Aphasia, alexia and agraphia. New York: Churchill Livingstone. |
[16] |
Bogen JE, Bogen GM (1976) Wernicke’s Region–where is it? Ann N Y Acad Sci 280: 834-843. doi: 10.1111/j.1749-6632.1976.tb25546.x
![]() |
[17] |
DeWitt I, Rauschecker JP (2013) Wernicke’s area revisited: parallel streams and word processing. Brain Lang 127: 181-191. doi: 10.1016/j.bandl.2013.09.014
![]() |
[18] | Dronkers NF, Redfern BB, Knight RT (2000) The neural architecture of language disorders. New Cogn Neurosci 2: 949-958. |
[19] |
Dronkers NF, Plaisant O, Iba-Zizen MT, Cabanis EA (2007) Paul Broca’s historic cases: high resolution MR imaging of the brains of Leborgne and Lelong. Brain 130: 1432-1441. doi: 10.1093/brain/awm042
![]() |
[20] |
Grodzinsky Y, Santi A (2008) The battle for Broca’s region. Trends Cogn Sci 12: 474-480. doi: 10.1016/j.tics.2008.09.001
![]() |
[21] | Bernal B, Ardila A, Rosselli M (2015) Broca’s area network in language function: A pooling-data connectivity study. Front Psycho 6: 687. |
[22] |
Ardila A, Bernal B, Rosselli M (2014) Participation of the insula in language revisited: A meta-analytic connectivity study. J Neuroling 29: 31-41. doi: 10.1016/j.jneuroling.2014.02.001
![]() |
[23] | Ardila A, Bernal B, Rosselli M (2015) Language and visual perception associations: Meta-analytic connectivity modeling of Brodmann area 37. Behav Neurol 2015: 565871 |
[24] | Ardila A, Bernal B, Rosselli M (2016) How localized are language brain areas? A review of Brodmann areas involvement in language. Arch Clin Neuropsychol 31: 112-122. |
[25] | Brannon EM, Cabeza R, Huettel SA, et al. (2008) Principles of cognitive. Neuroscience 3: 757. Sunderland, MA: Sinauer Associates. |
[26] |
Pulvermüller F, Fadiga L (2010) Active perception: sensorimotor circuits as a cortical basis for language. Nature Rev Neurosci 11: 351-360. doi: 10.1038/nrn2811
![]() |
[27] | Clark J, Yallop C (1995) An Introduction to Phonetics and Phonology. Blackwell |
[28] | Cammoun L, Thiran J P, Griffa A, et al. (2014) Intrahemispheric cortico-cortical connections of the human auditory cortex. Brain Struct Funct 220: 3537-3553. |
[29] |
Upadhyay J, Silver A, Knaus TA, et al. (2008) Effective and structural connectivity in the human auditory cortex. J Neurosc 28: 3341-3349. doi: 10.1523/JNEUROSCI.4434-07.2008
![]() |
[30] |
Bitan T, Lifshitz A, Breznitz Z, et al. (2010) Bidirectional connectivity between hemispheres occurs at multiple levels in language processing but depends on sex. J Neurosci 30: 11576-11585. doi: 10.1523/JNEUROSCI.1245-10.2010
![]() |
[31] |
Patel RS, Bowman FD, Rilling JK (2006) Determining hierarchical functional networks from auditory stimuli fMRI. Hum Brain Map 27: 462-470. doi: 10.1002/hbm.20245
![]() |
[32] |
Eckert MA, Kamdar NV, Chang CE, et al. (2008) A cross‐modal system linking primary auditory and visual cortices: Evidence from intrinsic fMRI connectivity analysis. Hum Brain Map 29: 848-857. doi: 10.1002/hbm.20560
![]() |
[33] |
Shams L, Ma WJ, Beierholm U (2005) Sound-induced flash illusion as an optimal percept. Neuroreport 16: 1923-1927. doi: 10.1097/01.wnr.0000187634.68504.bb
![]() |
[34] |
Balz J, Keil J, Romero YR, et al. (2016) GABA concentration in superior temporal sulcus predicts gamma power and perception in the sound-induced flash illusion. NeuroImage 125: 724-730. doi: 10.1016/j.neuroimage.2015.10.087
![]() |
[35] |
Bilecen D, Scheffler K, Schmid N, et al. (1998) Tonotopic organization of the human auditory cortex as detected by BOLD-FMRI. Hear Res 126: 19-27. doi: 10.1016/S0378-5955(98)00139-7
![]() |
[36] |
Da Costa S, Saenz M, Clarke S, et al. (2015) Tonotopic gradients in human primary auditory cortex: concurring evidence from high-resolution 7 T and 3 T fMRI. Brain Top 28: 66-69. doi: 10.1007/s10548-014-0388-0
![]() |
[37] |
Humphries C, Liebenthal E, Binder JR (2010) Tonotopic organization of human auditory cortex. Neuroimage 50: 1202-1211. doi: 10.1016/j.neuroimage.2010.01.046
![]() |
[38] |
Dorsaint-Pierre R, Penhune VB, Watkins KE, et al. (2006) Asymmetries of the planum temporale and Heschl’s gyrus: relationship to language lateralization. Brain 129: 1164-1176. doi: 10.1093/brain/awl055
![]() |
[39] | Zatorre RJ (2001) Neural specializations for tonal processing. Ann N Y Acad Sci 930: 193-210. |
[40] |
Zatorre RJ, Belin P (2001) Spectral and temporal processing in human auditory cortex. Ce Cortex 11: 946-953. doi: 10.1093/cercor/11.10.946
![]() |
[41] |
Zatorre RJ, Belin P, Penhune VB (2002) Structure and function of auditory cortex: music and speech. Trends Cogn Scien 6: 37-46. doi: 10.1016/S1364-6613(00)01816-7
![]() |
[42] | Schomers MR, Pulvermüller F (2016) Is the sensorimotor cortex relevant for speech perception and understanding? An integrative review. Front Hum Neurosc 10. |
[43] | Glick H, Sharma A (2016) Cross-modal Plasticity in Developmental and Age-Related Hearing Loss: Clinical Implications. Hear Res. |
[44] | Specht K, Baumgartner F, Stadler J, et al. (2014). Functional asymmetry and effective connectivity of the auditory system during speech perception is modulated by the place of articulation of the consonant-A 7T fMRI study. Front Psychol 5. |
[45] |
Menéndez-Colino LM, Falcón C, Traserra J, et al. (2007) Activation patterns of the primary auditory cortex in normal-hearing subjects: a functional magnetic resonance imaging study. Acta oto-laryngol 127: 1283-1291. doi: 10.1080/00016480701258705
![]() |
[46] |
Lasota KJ, Ulmer JL, Firszt JB, et al. (2003) Intensity-dependent activation of the primary auditory cortex in functional magnetic resonance imaging. J Comp Assist Tom 27: 213-218. doi: 10.1097/00004728-200303000-00018
![]() |
[47] | Yetkin FZ, Wolberg SC, Temlett JA, et al. (1990) Pure word deafness. South Afric Med J 78: 668-670. |
[48] |
Stefanatos GA, Joe WQ, Aguirre GK, et al. (2008) Activation of human auditory cortex during speech perception: effects of monaural, binaural, and dichotic presentation. Neuropsychologia 46: 301-315. doi: 10.1016/j.neuropsychologia.2007.07.008
![]() |
[49] |
Alain C, Reinke K, McDonald KL, et al. (2005) Left thalamo-cortical network implicated in successful speech separation and identification. Neuroimage 26: 592-599. doi: 10.1016/j.neuroimage.2005.02.006
![]() |
[50] |
Liebenthal E, Binder JR, Spitzer SM et al. (2005) Neural substrates of phonemic perception. Cereb Cortex 15: 1621-1631. doi: 10.1093/cercor/bhi040
![]() |
[51] |
Hall DA, Johnsrude IS, Haggard MP, et al. (2002) Spectral and temporal processing in human auditory cortex. Cereb Cortex 12: 140-149. doi: 10.1093/cercor/12.2.140
![]() |
[52] |
Hart HC, Hall DA, Palmer AR (2003) The sound-level-dependent growth in the extent of fMRI activation in Heschl’s gyrus is different for low-and high-frequency tones. Hear Res 179: 104-112. doi: 10.1016/S0378-5955(03)00100-X
![]() |
[53] |
Patterson RD, Uppenkamp S, Johnsrude IS, et al. (2002) The processing of temporal pitch and melody information in auditory cortex. Neuron 36: 767-776. doi: 10.1016/S0896-6273(02)01060-7
![]() |
[54] |
Obleser J, Boecker H, Drzezga A, et al. (2006) Vowel sound extraction in anterior superior temporal cortex. Hum Brain Map 27: 562-571. doi: 10.1002/hbm.20201
![]() |
[55] |
Izumi S, Itoh K, Matsuzawa H, et al. (2011) Functional asymmetry in primary auditory cortex for processing musical sounds: temporal pattern analysis of fMRI time series. Neuroreport 22: 470-473. doi: 10.1097/WNR.0b013e3283475828
![]() |
[56] |
Da Costa S, van der Zwaag W, Miller LM et al. (2013) Tuning in to sound: frequency-selective attentional filter in human primary auditory cortex. J Neurosc 33: 1858-1863. doi: 10.1523/JNEUROSCI.4405-12.2013
![]() |
[57] |
Calvert G, Campbell R (2003) Reading speech from still and moving faces: the neural substrates of visible speech. Cogn Neurosci J 15: 57-70. doi: 10.1162/089892903321107828
![]() |
[58] |
Wiegand K, Gutschalk A (2012) Correlates of perceptual awareness in human primary auditory cortex revealed by an informational masking experiment. Neuroimage 61: 62-69. doi: 10.1016/j.neuroimage.2012.02.067
![]() |
[59] | Benoit MM, Raij T, Lin FH, et al. (2010) Primary and multisensory cortical activity is correlated with audiovisual percepts. Hum Brain Map 31: 526-538. |
[60] |
Friederici AD, Meyer M, von Cramon DY (2000) Auditory language comprehension: an event-related fMRI study on the processing of syntactic and lexical information. Brain Lang 74: 289-300. doi: 10.1006/brln.2000.2313
![]() |
[61] | Deschamps I, Tremblay P (2014) Sequencing at the syllabic and supra-syllabic levels during speech perception: an fMRI study. Front Hum Neurosci 8. |
[62] |
Stoppelman N, Harpaz T, Ben‐Shachar M (2013) Do not throw out the baby with the bath water: choosing an effective baseline for a functional localizer of speech processing. Brain Behav 3: 211-222. doi: 10.1002/brb3.129
![]() |
[63] |
Belin P, Zatorre RJ, Ahad P (2002) Human temporal-lobe response to vocal sounds. Cogn Brain Res 13: 17-26. doi: 10.1016/S0926-6410(01)00084-2
![]() |
[64] | Conant LL, Liebenthal E, Desai A, et al. (2014) FMRI of phonemic perception and its relationship to reading development in elementary-to middle-school-age children. Neuroimage 89: 192-202. |
[65] |
Ardila A (1993) Toward a model of phoneme perception. Int J Neurosc 70: 1-12. doi: 10.3109/00207459309000556
![]() |
[66] | Nygaard C, Pisoni DB (1995) Speech Perception: New Directions in Research and Theory. In JL Miller, PD Eimas. Handbook of Perception and Cognition: Speech, Language, and Communication. San Diego: Academic Press. |
[67] |
Lin CY, Wang HC (2011) Automatic estimation of voice onset time for word-initial stops by applying random forest to onset detection. J Acooust Soc Amer 130: 514-525. doi: 10.1121/1.3592233
![]() |
[68] |
Zaehle T, Jancke L, Meyer M (2007) Electrical brain imaging evidences left auditory cortex involvement in speech and non-speech discrimination based on temporal features. Behav Brain Funct 3: 63. doi: 10.1186/1744-9081-3-63
![]() |
[69] |
Boatman D, Hart J, Lesser RP, et al. (1998) Right hemisphere speech perception revealed by amobarbital injection and electrical interference. Neurology 51: 458-464. doi: 10.1212/WNL.51.2.458
![]() |
[70] |
Specht K (2014) Neuronal basis of speech comprehension. Hear Res 307: 121-135. doi: 10.1016/j.heares.2013.09.011
![]() |
[71] | Basso A (2003) Aphasia and its therapy. New York: Oxford University Press. |
[72] |
Ardila, A (2010) A proposed reinterpretation and reclassification of aphasic syndromes. Aphasiology 24: 363-394. doi: 10.1080/02687030802553704
![]() |
[73] | Benson DF, Ardila A (1996) Aphasia: A clinical perspective. New York: Oxford University Press. |
[74] |
Gandour J, Dzemidzic M, Wong D, et al. (2003) Temporal integration of speech prosody is shaped by language experience: An fMRI study. Brain Lang 84: 318-336. doi: 10.1016/S0093-934X(02)00505-9
![]() |
[75] |
Ulrich G (1978) Interhemispheric functional relationships in auditory agnosia: an analysis of the preconditions and a conceptual model. Brain Lang 5: 286-300. doi: 10.1016/0093-934X(78)90027-5
![]() |
[76] | Ackermann H, Mathiak K (1999) Symptomatologie, pathologischanatomische Grundlaqen und Pathomechanismen zentraler Hörstörungen (reine Worttaubheit, auditive Agnosie, Rindentaubheit). Fortschritte Neurole·Psychiatr 67: 509-523. |
[77] |
Poeppel D (2001) Pure word deafness and the bilateral processing of the speech code. Cogn Sci 25: 679-693. doi: 10.1207/s15516709cog2505_3
![]() |
[78] | Takahashi N, Kawamura M, Shinotou H. et al. (1992) Pure word deafness due to left hemisphere damage. Cortex 28: 295-303. |
[79] | Kussmaul A (1877) Disturbances of speech. In H. von Ziemssen (Ed.), Enyclopedia of the practice of medicine. New York: William Wood. |
[80] | Auerbach SH, Allard T, Naeser M, et al. (1982) Pure word deafness. Analysis of a case with bilateral lesions and a defect at the prephonemic level. Brain 105: 271-300. |
[81] |
Feldmann H (2004) 200 years testing hearing disorders with speech, 50 years German speech audiometry—a review. Laryngo-rhino-otologie 83: 735-742. doi: 10.1055/s-2004-825717
![]() |
[82] |
Brody RM, Nicholas BD, Wolf MJ, et al. (2013) Cortical deafness: a case report and review of the literature. Otol Neurotol 34: 1226-1229. doi: 10.1097/MAO.0b013e31829763c4
![]() |
[83] |
Caramazza A, Berndt RS, Basili AG (1983) The selective impairment of phonological processing: A case study. Brain Lang 18: 128-174. doi: 10.1016/0093-934X(83)90011-1
![]() |
[84] |
Molfese DL, Molfese VJ, Espy KA (1999) The predictive use of event-related potentials in language development and the treatment of language disorders. Dev Neuropsychol 16: 373-377. doi: 10.1207/S15326942DN1603_19
![]() |
[85] |
Eimas PD (1999) Segmental and syllabic representations in the perception of speech by young infants. J Acoust Soc Amer 105: 1901-1911. doi: 10.1121/1.426726
![]() |
[86] |
Vannest JJ, Karunanayaka PR, Altaye M, et al. (2009) Comparison of fMRI data from passive listening and active—response story processing tasks in children. J Magn Res Imaging 29: 971-976. doi: 10.1002/jmri.21694
![]() |
[87] |
Ashtari M, Lencz T, Zuffante P, et al. (2004) Left middle temporal gyrus activation during a phonemic discrimination task. Neuroreport 15: 389-393. doi: 10.1097/00001756-200403010-00001
![]() |
[88] |
Dean III DC, Dirks H, O’Muircheartaigh J, et al. (2014) Pediatric neuroimaging using magnetic resonance imaging during non-sedated sleep. Ped Radiol 44: 64-72. doi: 10.1007/s00247-013-2752-8
![]() |
[89] |
Dehaene-Lambertz G, Dehaene S, Hertz-Pannier L (2002) Functional neuroimaging of speech perception in infants. Science 298: 2013-2015. doi: 10.1126/science.1077066
![]() |
[90] |
Altman NR, Bernal B (2001) Brain Activation in Sedated Children: Auditory and Visual Functional MR Imaging 1. Radiology 221: 56-63. doi: 10.1148/radiol.2211010074
![]() |
[91] |
Gemma M, de Vitis A, Baldoli C, et al. (2009) Functional magnetic resonance imaging (fMRI) in children sedated with propofol or midazolam. J Neurosurg Anesthesiol 21: 253-258. doi: 10.1097/ANA.0b013e3181a7181d
![]() |
[92] |
Dobrunz UEG, Jaeger K, Vetter G (2007) Memory priming during light anaesthesia with desflurane and remifentanil anaesthesia. Br J Anaesthesia 98: 491-496. doi: 10.1093/bja/aem008
![]() |
[93] |
Ford JM, Dierks T, Fisher DJ, et al. (2012) Neurophysiological studies of auditory verbal hallucinations. Schiz Bull 38: 715-723. www.fmriconsulting.com/brodmannconn/index.php?q=BA_41 doi: 10.1093/schbul/sbs009
![]() |
[94] |
Taylor RL, Campbell GT (1976) Sensory interaction: Vision is modulated by hearing. Perception 5: 467. doi: 10.1068/p050467
![]() |
[95] |
Poirier C, Collignon O, DeVolder AG, et al. (2005) Specific activation of the V5 brain area by auditory motion processing: an fMRI study. Cogn Brain Res 25: 650-658. doi: 10.1016/j.cogbrainres.2005.08.015
![]() |
[96] |
Poirier C, Collignon O, Scheiber C, et al. (2006) Auditory motion perception activates visual motion areas in early blind subjects. Neuroimage 31: 279-285. doi: 10.1016/j.neuroimage.2005.11.036
![]() |
[97] |
Collignon O, Dormal G, Albouy G, et al. (2013) Impact of blindness onset on the functional organization and the connectivity of the occipital cortex. Brain 136: 2769-2783. doi: 10.1093/brain/awt176
![]() |
[98] |
De Volder AG, Toyama H, Kimura Y, et al. (2001) Auditory triggered mental imagery of shape involves visual association areas in early blind humans. Neuroimage 14: 129-139. doi: 10.1006/nimg.2001.0782
![]() |
[99] |
Finney EM, Fine I, Dobkins KR (2001) Visual stimuli activate auditory cortex in the deaf. Nature Neurosci 4: 1171-1173. doi: 10.1038/nn763
![]() |
[100] |
Shiell MM, Champoux F, Zatorre RJ (2015) Reorganization of auditory cortex in early-deaf people: Functional connectivity and relationship to hearing aid use. J Cogn Neurosci 27: 150-163. doi: 10.1162/jocn_a_00683
![]() |
[101] | Flemming ES (2013) Auditory representations in phonology. Routledge. |
[102] | McGurk H, MacDonald J (1976) Hearing lips and seeing voices. Nature 264: 746-748. |
[103] |
Burton MW, LoCasto PC, Krebs-Noble D, et al. (2005) A systematic investigation of the functional neuroanatomy of auditory and visual phonological processing. Neuroimage 26: 647-661. doi: 10.1016/j.neuroimage.2005.02.024
![]() |
[104] |
Longoni F, Grande M, Hendrich V, et al. (2005) An fMRI study on conceptual, grammatical, and morpho-phonological processing. Brain Cogn 57: 131-134. doi: 10.1016/j.bandc.2004.08.032
![]() |
[105] | Cammoun L, Thiran JP, Griffa A, et al. (2014) Intrahemispheric cortico-cortical connections of the human auditory cortex. Brain Struct Function: 1-17. |
[106] |
Upadhyay J, Ducros M, Knaus TA, et al. (2007) Function and connectivity in human primary auditory cortex: a combined fMRI and DTI study at 3 Tesla. Cereb Cortex 17: 2420-2432. doi: 10.1093/cercor/bhl150
![]() |
[107] |
Saur D, Kreher BW, Schnell S, et al. (2008) Ventral and dorsal pathways for language. Proc Natl Acad Sci U S A 105: 18035-18040. doi: 10.1073/pnas.0805234105
![]() |
[108] |
Ardila A (2011) There are two different language systems in the brain. J Behav Brain Sci 1: 23. doi: 10.4236/jbbs.2011.12005
![]() |
[109] |
Ardila A (2012) Interaction between lexical and grammatical language systems in the brain. Phys Life Rev 9: 198-214. doi: 10.1016/j.plrev.2012.05.001
![]() |
[110] |
Duffau H, Gatignol P, Mandonnet E, et al. (2008) Intraoperative subcortical stimulation mapping of language pathways in a consecutive series of 115 patients with Grade II glioma in the left dominant hemisphere. J Neurosurg 109: 461-471. doi: 10.3171/JNS/2008/109/9/0461
![]() |
[111] |
Bernal B, Ardila A (2009) The role of the arcuate fasciculus in conduction aphasia. Brain 132: 2309-2316. doi: 10.1093/brain/awp206
![]() |
[112] | Brewer AA, Barton B (2016) Maps of the Auditory Cortex. An Rev Neurosci 39: 385-407. |
[113] | Manca AD, Grimaldi M (2016) Vowels and Consonants in the Brain: Evidence from Magnetoencephalographic Studies on the N1m in Normal-Hearing Listeners. Front Psychol 7. |
[114] |
Caclin A, Fonlupt P (2006) Functional and effective connectivity in an Fmri study of an auditory‐related task. Eur J Neurosc 23: 2531-2537. doi: 10.1111/j.1460-9568.2006.04773.x
![]() |
[115] |
Binder JR, Frost JA, Hammeke TA et al. (2000) Human temporal lobe activation by speech and nonspeech sounds. Cereb Cortex 10: 512-528. doi: 10.1093/cercor/10.5.512
![]() |
[116] | Hickok G, Poeppel D (2007) The cortical organization of speech processing. Nat Rev Neurosci 8: 393-402. |
[117] |
Vigneau M, Beaucousin V, Herve PY, et al. (2006) Meta-analyzing left hemisphere language areas: phonology, semantics, and sentence processing. Neuroimage 30: 1414-1432. doi: 10.1016/j.neuroimage.2005.11.002
![]() |
[118] |
Tervaniemi M, Hugdahl K (2003) Lateralization of auditory-cortex functions. Brain Res Rev 43: 231-246. doi: 10.1016/j.brainresrev.2003.08.004
![]() |
[119] |
Zhang L, Xi J, Xu G, et al. (2011) Cortical dynamics of acoustic and phonological processing in speech perception. PloS one 6: e20963. doi: 10.1371/journal.pone.0020963
![]() |
[120] |
Bernal B, Ardila A (2014) Bilateral representation of language: A critical review and analysis of some unusual cases. J Neuroling 28: 63-80. doi: 10.1016/j.jneuroling.2013.10.002
![]() |