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Jordan left derivations in full and upper triangular matrix rings

December 2010
Electronic Journal of Linear Algebra 20(1)

Authors:

Jilin University

Left derivations and Jordan left derivations in full and upper triangular matrix rings over unital associative rings are characterized.

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Electronic Journal of Linear Algebra

Volume 20 ELA Volume 20 (2010) Article 54

2010

Jordan le derivations in full and upper triangular

matrix rings

Xiao-Wei Xu

xuxw@jlu.edu.cn

Hong-Ying Zhang

Follow this and additional works at: h?p://repository.uwyo.edu/ela

=is Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of

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Recommended Citation

Xu, Xiao-Wei and Zhang, Hong-Ying. (2010), "Jordan le> derivations in full and upper triangular matrix rings", Electronic Journal of

Linear Algebra, Volume 20.

DOI: h?p://dx.doi.org/10.13001/1081-3810.1407

ELA

JORDAN LEFT DERIVATIONS IN FULL AND UPPER

TRIANGULAR MATRIX RINGS∗

XIAO WEI XU†AND HONG YING ZHANG†

Abstract. In this paper, left derivations and Jordan left derivations in full and upper triangular

matrix rings over unital associative rings are characterized.

Key words. Left derivations, Jordan left derivations, Full matrix rings, Triangular matrix rings.

AMS subject classiﬁcations. 16S50, 16W25.

1. Introduction. Let Rbe an associative ring. An additive mapping δ:R→M

from Rinto a bimodule RMRis called a module derivation if δ(xy) = δ(x)y+xδ(y)

holds for all x, y ∈R. Particularly, the module derivation from Rinto its regular

bimodule RRRis well known as the ring derivation (usually called derivation). Ob-

viously, the concept of module derivations depends heavily on the bimodule structure

of M, i.e., if Mis a left R-module but not a right R-module, this concept will not

happen. However, a small modiﬁcation can lead a new concept, that is, the concept

of module left derivations. Exactly, an additive mapping δfrom a ring Rinto its left

module RMis called a module left derivation if δ(xy) = xδ(y) + yδ(x) holds for all

x, y ∈R. Particularly, a module left derivation from Rinto its left regular module

RRis called a ring left derivation (usually called a left derivation ).

The concept of (module) left derivations appeared in Breˇsar and Vukman [8] at

ﬁrst. They obtained that a left derivation in a prime ring must be zero, that a left

derivation in a semiprime ring must be a derivation such that its range is contained

in the center, and that a continuous linear left derivation in a Banach algebra A

must map Ainto its Jacobson radical Rad(A). Since left derivations act in accord

with derivations in a commutative ring, the result on Banach algebra by Breˇsar and

Vukman can be seen as a generalization of the one by Singer and Wermer [22] which

states that a continuous linear derivation in a commutative Banach algebra Amust

map Ainto its Jacobson radical Rad(A).

Since Breˇsar and Vukman initiated the study of left derivations in noncom-

∗Received by the editors on April 20, 2010. Accepted for publication on October 22, 2010.

Handling Editor: Robert Guralnick.

†College of Mathematics, Jilin University, Changchun 130012, PR China (xuxw@jlu.edu.cn,

zlzxzhy@163.com). Supported by the NNSF of China (No. 10871023 and No. 11071097), 211

Project, 985 Project and the Basic Foundation for Science Research from Jilin University.

753

Electronic Journal of Linear Algebra ISSN 1081-3810

A publication of the International Linear Algebra Society

Volume 20, pp. 753-759, December 2010

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ELA

754 X.W. Xu and H.Y. Zhang

mutative rings, many related results have appeared for both Banach algebras (for

example, see [12, 14, 15, 16, 20, 21, 23, 24]) and prime rings (for example, see

[1, 3, 4, 5, 10, 13, 24, 25, 26]). However, in this paper, we will concerned ourselves

with (Jordan) left derivations in full and upper triangular matrix rings over unital

associative rings.

Recall that an additive mapping δ:R→Mfrom a ring Rinto its bimodule RMR

is called a module Jordan derivation if δ(x2) = δ(x)x+xδ(x) holds for all x∈R.

Particularly, a module Jordan derivation from Rinto its regular bimodule RRRis

called a ring Jordan derivation (usually called a Jordan derivation). Similarly, an

additive mapping δfrom a ring Rinto its left module RMis called a module Jordan

left derivation if δ(x2) = 2xδ(x) holds for all x∈R. Particularly, a module Jordan

left derivation from Rinto its left regular module RRis called a ring Jordan left

derivation (usually called a Jordan left derivation). For both Banach algebras and

prime rings, Jordan left derivations have been studied broadly.

On the other hand, (Jordan) derivations in full and upper triangular matrix rings

over unital rings have been characterized (see [2, 6, 7, 9, 17, 18, 19]). This short note

will characterize (Jordan) left derivations in full and upper triangular matrix rings

over unital rings.

Unless stated otherwise, Ralways denotes a unital associative ring with left R-

module RV. Let Mn(R) and Tn(R) be the full and upper triangular matrix ring over

Rseparately. In a natural fashion, Mn(V), the set of all n×nmatrices over V, is a

left Mn(R) module. Similarly, Tn(V), the set of all n×nupper triangular matrices

over V, is a left Tn(R) module. The symbol eij , 1 ≤i, j ≤n, will be used for a

matrix having all entries zero except the (i, j)-entry which is equal to 1. Note that

for a module Jordan left derivation µ:R→V,µ(x2) = 0 holds for all x∈Rif and

only if 2µ(x) = 0 holds for all x∈R. The “if” part is obvious. And for all x∈R,

2µ(x) = µ(2x) = µ(x2+ 2x+ 12) = µ((x+ 1)2) = 0

proves the other part. For convenience, a module Jordan left derivation µ:R→Vis

called strong if µ(x2) = 2µ(x) = 0 holds for all x∈R. And so, a module Jordan left

derivation µ:R→Vis strong if and only if µ(V)⊆ {x∈V|2x= 0}. Particularly,

every module Jordan left derivation is strong when Vis 2-torsion. And the unique

strong module Jordan left derivation is zero when Vis 2-torsion free.

Now we record some basic facts on module (Jordan) left derivations as following.

Remark 1.1. Let µ:R→Vbe a module Jordan left derivation. Then µ(e) = 0

for all e=e2∈R.

Proof. By µ(e) = µ(e2) = 2eµ(e), we have that eµ(e) = e(2eµ(e)) = 2eµ(e).

Hence eµ(e) = 0, and then µ(e) = 2eµ(e) = 0.

Electronic Journal of Linear Algebra ISSN 1081-3810

A publication of the International Linear Algebra Society

Volume 20, pp. 753-759, December 2010

http://math.technion.ac.il/iic/ela

ELA

Jordan Left Derivations in Full and Upper Triangular Matrix Rings 755

Remark 1.2. Let µ:Mn(R)→Mn(V) (resp., µ:Tn(R)→Tn(V)) be a module

Jordan left derivation. Then µ(eii ) = 0 for all 1 ≤i≤n, and µ(xeij ) = 0 for all

x∈Rand for all i6=j(resp., i < j).

Proof. By Remark 1.1, we have µ(eii) = 0 for all 1 ≤i≤nand µ(eii +xeij ) = 0

for all x∈Rand for all i6=j(resp., i < j). Hence, µ(xeij ) = µ(eii +xeij)−µ(eii ) = 0

for all i6=j(resp., i < j).

Remark 1.3. Let µ:R→Vbe a module left derivation. Then xy −yx ∈ker µ

for all x, y ∈R.

Proof. It can be proved by direct checking.

Remark 1.4. Let µ:R→Vbe a strong module Jordan left derivation. Then

µ(xy +yx) = 0 for all x, y ∈R.

Proof. For all x, y ∈R,µ(xy +yx) = µ(x2+y2+xy +yx) = µ((x+y)2) = 0.

2. Main results. Firstly, we characterize module left derivations in full and

upper triangular matrix rings over unital associative rings.

Proposition 2.1. For n≥2, a module left derivation µ:Mn(R)→Mn(V)

must be zero.

Proof. By Remark 1.2, µ(xeij ) = 0 for all i6=jand for all x∈R. On the

other hand, for all i6=jand for all x∈R,µ(xeii) = µ((xeij )ej i ) = (xeij )µ(eji ) +

eji µ(xeij ) = 0 which completes the proof.

Proposition 2.2. For n≥2, a mapping µ:Tn(R)→Tn(V)is a module left

derivation if and only if there exist module left derivations µi:R→V(1 ≤i≤n)

such that for all A= (aij )∈Tn(R),

µ:







a11 a12 ··· a1n

a22 ··· a1n

....

ann







7→







µ1(a11)µ2(a11 )··· µn(a11)

0··· 0

....







Proof. We merely deal with the “only if” part since the other part can be checked

directly. By Remark 1.2, we have µ(eii) = 0 for all 1 ≤i≤nand µ(xeij ) = 0

for all i < j and for all x∈R. For all x∈Rand for all 1 ≤i≤n,µ(xeii) =

µ(eii(xeii )) = eiiµ(xeii ). Particularly, for all x∈Rand for all 2 ≤i≤n, 0 =

µ(xe1i) = µ(e1i(xeii)) = e1iµ(xeii ). Hence, µ(xeii) = 0 for all x∈Rand for all

2≤i≤nsince µ(xeii) = eii µ(xeii). For each 1 ≤i≤n, let µi:R→Vbe

the mapping such that µi(x) is the (1, i)-entry of µ(xe11) for all x∈R. Obviously,

Electronic Journal of Linear Algebra ISSN 1081-3810

A publication of the International Linear Algebra Society

Volume 20, pp. 753-759, December 2010

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ELA

756 X.W. Xu and H.Y. Zhang

each µiis an additive mapping. Moreover, for all x, y ∈R,µi(xy) is the (1, i)-entry

of µ(xye11 ) = xe11µ(ye11) + ye11 µ(xe11) for all 1 ≤i≤n. And so, for each µi,

µi(xy) = xµi(y) + yµi(x) holds for all x∈R, which completes the proof.

By Proposition 2.2, we have the following corollaries.

Corollary 2.3. For n≥2, there exist nonzero module left derivations from

Tn(R)into Tn(V)if and only if there exist nonzero module left derivations from R

into V.

Corollary 2.4. Let Vbe an R-bimodule and n≥2. Then a module left

derivation µ:Tn(R)→Tn(V)which is also a module derivation must be zero.

If a (resp., module) left derivation is not a (resp., module) derivation, we call it

nontrivial or proper. By Proposition 2.2, we can construct some nontrivial examples

of (module) left derivations.

Example 2.5. Let R=Q[x]. Then for n≥2, a left derivation µof Tn(R) must

be the following form

µ:







a11(x)a12 (x)··· a1n(x)

a22(x)··· a1n(x)

....

ann(x)







7→







f1(x)a′

11(x)f2(x)a′

11(x)··· fn(x)a′

11(x)

0··· 0

....







where f1(x), f2(x),...,fn(x) are ﬁxed polynomials in Q[x].

Now we characterize module Jordan left derivations in full and upper triangular

matrix rings over unital associative rings.

Theorem 2.6. For n≥2, a mapping µ:Mn(R)→Mn(V)is a module Jordan

left derivation if and only if there exist strong module Jordan left derivations µij :

R→V(1 ≤i, j ≤n)such that for all A= (aij )∈Mn(R),µ(A) = (µij (trA)),

where trA =Pn

i=1 aii is the trace of A. Particularly the unique module Jordan left

derivation µ:Mn(R)→Mn(V)is zero when Vis 2-torsion free.

Proof. For the “if” part, we can obtain the conclusion by Remark 1.4 and the

fact that tr(A2) = Pn

i=1 a2

ii +Pi6=j(aij aji +ajiaij ). Now we deal with the “only if”

part. By Remark 1.2, we have µ(eii) = 0 for all 1 ≤i≤nand µ(xeij ) = 0 for all

Electronic Journal of Linear Algebra ISSN 1081-3810

A publication of the International Linear Algebra Society

Volume 20, pp. 753-759, December 2010

http://math.technion.ac.il/iic/ela

ELA

Jordan Left Derivations in Full and Upper Triangular Matrix Rings 757

i6=jand for all x∈R. For all i6=jand for all x∈R,

µ(xeii +xejj ) = µ((eij +xeji )2) = 2(eij +xej i)µ(eij +xej i) = 0.

For each 1 ≤i≤n, using µ(eii) = 0, we have that

2xeiiµ(xeii ) = µ((xeii)2) = µ(((x−1)eii +I)2)

= 2((x−1)eii +I)µ((x−1)eii +I) = 2((x−1)eii +I)µ(xeii).

And so, 2(I−eii)µ(xeii ) = 0 for all x∈Rand for all 1 ≤i≤n. For some j6=i,

we have that 0 = 2(I−ej j )µ(xejj ) = 2(I−ej j )µ(xeii) since µ(xeii +xej j ) = 0.

Particularly, we have that 2eiiµ(xeii ) = 0. Hence, 2µ(xeii) = 0 for all x∈Rand

for all 1 ≤i≤n. And so, µ(x2eii ) = 2xeiiµ(xeii ) = 0 for all x∈Rand for all

1≤i≤n. In fact, for all i6=j, we have obtained µ(xeii) = µ(xejj ) for all x∈R.

Let µij :R→V(1 ≤i, j ≤n) be the mapping such that µij(x) is the (i, j )-entry of

µ(xe11) for all x∈R. Then µij :R→V(1 ≤i, j ≤n) are strong module Jordan left

derivations which completes the proof.

If a (resp., module) Jordan left derivation is not a (resp., module) left derivation,

we call it nontrivial or proper. By Theorem 2.6, we can construct some nontrivial

examples of (module) Jordan left derivations.

Example 2.7. Let R=Z2[x], and let fij (x)∈R(1 ≤i, j ≤n) be ﬁxed

polynomials. For n≥2, we obtain a nontrivial Jordan left derivation µ:Mn(R)→

Mn(R) as µ(A(x)) = trA(x)′(fij (x)).

Theorem 2.8. For n≥2, a mapping µ:Tn(R)→Tn(V)is a module Jordan

left derivation if and only if there exist module Jordan left derivations

µk

ij :R→V(1 ≤i≤n, i ≤j≤n, 1≤k≤n)

such that all µk

ij but µ1

1j(1 ≤j≤n)are strong and µ(A) = Pn

k=1(µk

ij (akk )) for all

A= (aij )∈Tn(R).

Proof. It can be checked directly for the necessary part. Now we deal with the

suﬃcient part. By Remark 1.2, we have µ(eii) = 0 for all 1 ≤i≤nand µ(xeij ) = 0

(i < j) for all x∈R. Let

µk

ij :R→V(1 ≤i≤n, i ≤j≤n, 1≤k≤n)

be the (i, j)-entry of µ(xekk) for each x∈R. Obviously, each µk

ij is an additive

mapping such that µ(A) = Pn

k=1(µk

ij (akk )) for all A= (aij )∈Tn(R). Now let xbe

an arbitrary element in R. For all 1 ≤i≤n, using µ(eii ) = 0, we have that

2xeiiµ(xeii ) = µ((xeii)2) = µ(((x−1)eii +I)2)

= 2((x−1)eii +I)µ((x−1)eii +I) = 2((x−1)eii +I)µ(xeii).

Electronic Journal of Linear Algebra ISSN 1081-3810

A publication of the International Linear Algebra Society

Volume 20, pp. 753-759, December 2010

http://math.technion.ac.il/iic/ela

ELA

758 X.W. Xu and H.Y. Zhang

And so, 2(I−eii)µ(xeii ) = 0 (1 ≤i≤n). This shows that 2µk

ij = 0 for all i6=k∈

{1,2,...,n}and for all i≤j≤n. Particularly, for all 2 ≤i≤n, using µ(e1i) = 0,

we have that

2xeiiµ(xeii ) = µ((xeii)2) = µ((xeii +e1i)2)

= 2(xeii +e1i)µ(xeii +e1i) = 2xeiiµ(xeii ) + 2e1iµ(xeii).

Hence, for all 2 ≤i≤n, we have 2e1iµ(xeii) = 0 which shows that 2eii µ(xeii ) = 0.

Thus, 2µk

ij = 0 for all 2 ≤k≤nand for all 1 ≤i≤j≤n. At the same time we have

proved that 2µ1

ij = 0 for all 2 ≤i≤nand for all i≤j≤n. All of these shows that

µk

ij (x2) = 0 for all 2 ≤k≤nand for all 1 ≤i≤j≤n, and that µ1

ij (x2) = 0 for all

2≤i≤nand for all i≤j≤n. So all µk

ij but µ1

1j(1 ≤j≤n) are strong module

Jordan left derivations. Moreover it can be checked directly that each µ1

1j(1 ≤j≤n)

is a module Jordan left derivation, which completes the proof.

By Theorem 2.8, we have:

Corollary 2.9. Let RVbe 2-torsion free. Then for n≥2, there exist proper

module Jordan left derivations from Tn(R)into Tn(V)if and only if there exist proper

module Jordan left derivations from Rinto V.

By Corollary 2.9 and known results on (left) derivations in 2-torsion free prime

rings [8, 11, 13], we have:

Corollary 2.10. Let Rbe 2-torsion free prime ring. Then there is not proper

module Jordan left derivations of Tn(R).

Acknowledgment. We wish to thank the referee who has clariﬁed the concept of

strong module Jordan left derivation, which is crucial to clear Jordan left derivations.

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http://math.technion.ac.il/iic/ela

ELA

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Dec 2008
HACET J MATH STAT

Let R be a prime ring. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R → R is called a generalized (θ, φ)-derivation if there exists a (θ, φ)-derivation d : R → R such that F(xy) = F(x)θ(y)+ φ(x)d(y) holds for all x, y ∈ R. Let J be a nonzero Jordan ideal of R. In the present paper we begin by proving the following: If F is a generalized (θ, φ)-derivation on R which acts as a homomorphism or as an anti- homomorphism on J, then either d = 0 or J ⊆ Z(R).

Jordan derivations and Jordan left derivations of Banach algebras

Article

Full-text available

Apr 2002

We show that every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the Jacobson radical of A, and hence every left derivation on a semisimple Banach algebra is always zero.

On Lie ideals and Jordan left derivation of prime rings

Article

Full-text available

Jan 2000

Let R be a 2-torsion free prime ring and let U be a Lie ideal of R such that u 2 ∈U for all u∈U. In the present paper it is shown that if d is an additive mapping of R into itself satisfying d(u 2 )=2ud(u) for all u∈U, then d(uv)=ud(v)+vd(u) for all u,v∈U.

Continuity of Lie derivations on Banach algebras

Article

Full-text available

Oct 1998
P EDINBURGH MATH SOC

The separating subspace of any Lie derivation on a semisimple Banach algebra A is contained in the centre of A.

A note on generalized left (Θ, Φ)-derivations in prime rings

Article

Aug 2011
HACET J MATH STAT

In this paper we describe generalized left (Θ, Φ)-derivations in prime rings, and prove that an additive mapping in a ring R acting as a homomorphism or anti-homomorphism on an additive subgroup S of R must be either a mapping acting as a homomorphism on S or a mapping acting as an anti-homomorphism on S, through which some related results are improved.

Jordan Derivations of Prime Rings

Article

Jan 1958

I.N. Herstein

Some results on Jordan left derivations in Banach algebras

Article

Jan 1999

Yong-Soo Jung

The purpose of this paper is to obtain Jordan left derivations that map into the Jacobson radical: (i) Let d be a spectrally bounded Jordan left derivation on a Banach algebra A. If [d(x),x]∈rad(A) for all x∈A, then d(A)⊆rad(A). (ii) Let d be a Jordan left derivation on a unital Banach algebra A with the condition sup{r(z -1 d(z))∣z∈Ainvertible}<∞. If [d(x),x]∈rad(A) for all x∈A, then d(A)⊆rad(A).

The Image of a Derivation is Contained in the Radical

Article

Nov 1988
ANN MATH

Marc P. Thomas

In 1955 I. M. Singer and J. Wermer proved that a bounded derivation on a commutative Banach algebra maps into the (Jacobson) radical; they conjectured that this result holds even if the derivation is unbounded. We give a proof of this conjecture. The central idea in the proof is the introduction of the concept of a recalcitrant system of elements in a commutative radical Banach algebra. Such systems put algebraic constraints upon a derivation which prevent the derivation from mapping outside of the radical.

On Left Derivations and Related Mappings

Article

Sep 1990

Let R be a ring and X be a left R-module. The purpose of this paper is to investigate additive mappings D1: R → X and D2: R → X that satisfy D1(ab) = aD1(b) + bD1(a), a, b ∈ R (left derivation) and D2(a2) = 2aD2(a), a ∈ R (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of R into X implies R is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.

Continuity of Derivations on Commutative Algebras

Article

Jan 1969

B. E. Johnson

Jordan left derivations in full and upper triangular matrix rings

Abstract

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