Math. J. Okayama Univ. 51 (2009), 111–119

ON LIE IDEALS AND LEFT JORDAN σ-CENTRALIZERS
OF 2-TORSION-FREE RINGS
Dedicated to Miguel Ferrero on his 70th. birthday
Wagner CORTES and Claus HAETINGER
Abstract. B. Zalar proved that any left (resp. right) Jordan centralizer on a 2-torsion-free semiprime ring is a left (resp. right) centralizer.
We prove this question changing the semiprimality condition on R.
The main result of this paper is the following. Let R be a 2-torsionfree ring which has a commutator right (resp. left) nonzero divisor and
let G: R → R be a left (resp. right) Jordan σ-centralizer mapping ofR, where σ is an automorphism of R. Then G is a left (resp. right)
σ-centralizer mapping of R.

1. Introduction
This research has been motivated by the works of J. Vukman [8], J. Vukman and I.K.-Ulbl [9] and B. Zalar [10].
Throughout the paper R will denote an associative ring, with center Z(R),not necessarily with an identity element, unless otherwise stated, and U
denotes a Lie ideal of R. As usual, [x, y] denotes the commutator xy − yx.
Recall that if R is a ring, R has a Lie structure by the bracket product[x, y], for x, y ∈ R. A Lie ideal of R is any additive subgroup U of R with[u, r] = ur − ru ∈ U for every u ∈ U and r ∈ R [7].
An additive mapping G: R → R is called a left (resp. right) centralizer,
if G(xy) = G(x)y (resp. G(xy) = xG(y)) holds for all x, y ∈ R. If a ∈ R,
then La (x) = ax is a left centralizer and Ra (x) = xa is a right centralizer.
Following B. Zalar [10], G is a centralizer mapping if G is both left and rightcentralizer.
If R is a ring with involution ⋆, then every additive mapping E: R → R
which satisfies E(x2 ) = E(x)x⋆ + xE(x) for all x ∈ R is called a Jordan⋆-derivation. Following [10], these mappings are closely connected with aquestion of representability of quadratic forms by bilinear forms. In [3,Mathematics Subject Classification. 16A12; 16A68; 16A72; 16N60; 16W25.
Key words and phrases. Left (Right) σ-Centralizer, Left (Right) Jordan σ-Centralizer,
Lie Ideals, Commutator.
The second named author was partially supported by Fundação de Amparo à Pesquisa
do Estado do Rio Grande do Sul (Fapergs, Brazil), and by Fundação Coordenação de
Aperfeiçoamento de Pessoal de Nı́vel Superior (CAPES, Brazil). Some results of thispaper were obtained when he was visiting the Universidade Federal do Rio Grande do Sul(UFRGS, Brazil) as a visitor researcher.
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proof of Theorem 2.1], M. Bres̆ar and B. Zalar obtained a representation of
Jordan ⋆-derivations in terms of left and right centralizers on the algebra ofcompact operators on a Hilbert space.
If we introduce a new product in an associative ring R, given by x ◦ y =xy + yx, then a Jordan derivation is an additive mapping d which satisfiesd(x◦y) = d(x)◦y + x◦d(y), for every x, y ∈ R, and a Jordan homomorphism
is an additive mapping h which satisfies h(x◦y) = h(x)◦h(y), for all x, y ∈ R.
Therefore, we can define a Jordan centralizer to be an additive mapping T
which satisfies T (x ◦ y) = T (x) ◦ y = x ◦ T (y), for every x, y ∈ R. Since theproduct ◦ is commutative, there is no difference between the left and right
Jordan centralizers.
An easy computation shows that every centralizer is also a Jordan centralizer. In [10], B. Zalar proves that every Jordan centralizer of a semiprime
ring is a centralizer.
Let G: R → R be an additive mapping and σ an endomorphism of R. We
call G a left (resp. right) Jordan σ-centralizer if G(x2 ) = G(x)σ(x) (resp.
G(x2 ) = σ(x)G(x)) holds for all x ∈ R. Similarly, if U is a Lie ideal of R,
then an additive mapping G: R → R is said to be a left (resp. right) Jordan
σ-centralizer of U into R in case that the above corresponding conditions
are satisfied for all x ∈ U . When σ = idR we have the usual well-known
definitions of left and right Jordan centralizer mappings.
There exists a type of Lie ideals that occurs in some works involving usual
derivations [2], namely square closed. A Lie ideal U of R is said to be squareclosed if it verifies u2 ∈ U , for every u ∈ U . It follows that for a squareclosed Lie ideal U of R, uv + vu ∈ U and 2uv ∈ U , for any u, v ∈ U [5, page251]. These remarks will be freely used in the paper.
In the whole paper, we will consider σ as an automorphism of a ring R.
2. Results
B. Zalar [10, Proposition 1.4] proved that any left (resp. right) Jordan
centralizer of a 2-torsion-free semiprime ring is a left (resp. right) centralizer.
It is our aim in this paper to prove the result above changing the semiprimality condition on R by the existence of a commutator right (resp. left)
nonzero divisor. This last condition seems to be artificial. Then, for thesake of completeness, we include at the end of the paper some examples ofrings that are not semiprime and having commutators nonzero divisors.
Furthermore, we state this result for Lie ideals, proving the following
Theorem 2.1. Let R be a 2-torsion-free ring, U a square closed Lie ideal of
R which has a commutator right (resp. left) nonzero divisor, and G: R → R
a left (resp. right) Jordan σ-centralizer mapping of U into R. Then G is aleft (resp. right) σ-centralizer mapping of U into R.

ON LIE IDEALS AND LEFT JORDAN σ-CENTRALIZERS

113

Since U = R is obviously a square closed Lie ideal of R, Theorem 2.1
is also true for left (resp. right) Jordan σ-centralizer mappings of R. We
include an example by M. Bres̆ar showing that the semiprimality and theright (resp. left) nonzero divisor commutator assumptions are independent
each other [4, Example].
Furthermore, if R is a 2-torsion-free ring, U is a square closed Lie idealof R, and G: R → R is a left (resp. right) σ-centralizer of U into R, thenan easy computation gives that G(xyz) = G(x)σ(y)σ(z) (resp. G(xyz) =σ(x)σ(y)G(z)), for every x, y, z ∈ U . A natural question is to ask whether
the converse is also true. We prove the following
Theorem 2.2. Let R be a 2-torsion-free ring, U a square closed Lie idealof R which has a commutator right (resp. left) nonzero divisor, and G:
R → R an additive mapping that satisfies G(xyz) = G(x)σ(y)σ(z) (resp.
G(xyz) = σ(x)σ(y)G(z)), for every x, y, z ∈ U . Then G is a left (resp.
right) σ-centralizer of U into R.
Finally, if G: R → R is a left and right σ-centralizer map of a ring R, weeasily have 2G(xyz) = G(x)σ(y)σ(z) + σ(x)σ(y)G(z), for each x, y, z ∈ R.
We obtained a result that provides a converse of this fact.
Theorem 2.3. Let R be a 2-torsion-free ring, U a square closed Lie ideal of
R which has an element that is nonzero divisor, and G: R → R an additive
mapping that satisfies 2G(xyz) = G(x)σ(y)σ(z) + σ(x)σ(y)G(z), for every
x, y, ∈ U . Then G is a σ-centralizer of U into R.
If there is no possibility of misunderstanding, we will consider the mappings G: R → R always as additive mappings.
3. Proofs
Before proving Theorem 2.1, let us point out that in case R has an identity
element, Theorem 2.1 can be easily proved for U = R. Namely, in this caseone puts x + 1 for x in G(x2 ) = G(x)σ(x) (resp. G(x2 ) = σ(x)G(x)),
where 1 denotes the identity element, which gives G(x) = G(1)σ(x) (resp.
G(x) = σ(x)G(1)). Thus G is left (resp. right) σ-centralizer.
Now let U be a Lie ideal of R. To facilitate our discussion we define amapping δl : R2 → R (resp. δr : R2 → R) such that δl (x, y) := G(xy) −G(x)σ(y) , for G: R → R a left Jordan σ-centralizer mapping of U intoR, (resp. δr (x, y) := G(xy) − σ(x)G(y), for G: R → R a right Jordan σcentralizer of U into R). It is easy to see that δl and δr are additive withrespect to both arguments. Moreover, if δl (resp. δr ) is zero, then G is a left
(resp. right) σ-centralizer mapping of U into R.
We need the following Lemma:

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Lemma 3.1. Let R be a 2-torsion-free ring, U a square closed Lie ideal of
R and G: R → R a left (resp. right) Jordan σ-centralizer mapping of U
into R. Then, for every a, b, c ∈ U , the following statements hold:(i). G(ab + ba) = G(a)σ(b) + G(b)σ(a) (resp. G(ab + ba) = σ(a)G(b) +σ(b)G(a));(ii). G(aba) = G(a)σ(b)σ(a) (resp. G(aba) = σ(a)σ(b)G(a));(iii). G(abc + cba) = G(a)σ(b)σ(c) + G(c)σ(b)σ(a) (resp. G(abc + cba) =σ(a)σ(b)G(c) + σ(c)σ(b)G(a));(iv). δl (a, b)σ([a, b]) = 0 (resp. σ([a, b])δr (a, b) = 0).
Proof. We only show the left Jordan σ-centralizer case, because the rightcase can be proved in an analogous way.
(i) and (ii) are easily obtained in the way similar to that in the proof of[5, Theorem 1.3]; while (iii) follows easily as in the proof of [4, Lemma 2.1(iii)].
Take now a, b ∈ U . Replacing c by 2ab in (iii), we get for β = 8ab(ab) +8(ab)ba,G(β) = 8G(ab(ab) + (ab)ba) = 8(G((ab)2 ) + G(ab2 a)).
Since R is 2-torsion-free, this shows (iv).

Note that in Lemma 3.1, item (i), we do not need the 2-torsion-free condition on R. Besides that, Lemma 3.1 holds in case σ is just an endomorphismof the ring R.
Proof of Theorem 2.1. We only show the left Jordan σ-centralizer case,because the right case is similar. By assumption, there exist elements a andb of U such that c[a, b] = 0 implies c = 0 for every c ∈ R. Since σ is an
automorphism of R, this implies that c · σ([a, b]) = 0 ⇒ c = 0. Then by
Lemma 3.1 (iv), we have(1)δl (a, b) = 0.

Our aim is to show that δl (x, y) = 0, for all x, y ∈ U . From Lemma 3.1(iv), we have(2)δl (x, y)σ([x, y]) = 0, for all x, y ∈ U.

Replacing x by x + a in (2), we get(3)δl (x, y)σ([a, y]) + δl (a, y)σ([x, y]) = 0, for all x, y ∈ U.

Now substitute y by y + b in (3). We get(4) δl (x, y)σ([a, b]) + δl (x, b)σ([a, y]) + δl (x, b)σ([a, b]) + δl (a, y)σ([x, b]) = 0,
for all x, y ∈ U .

ON LIE IDEALS AND LEFT JORDAN σ-CENTRALIZERS

115

Changing x by a in (4), using (1) and since R is a 2-torsion-free ring, we
obtain δl (a, y)σ([a, b]) = 0, for every y ∈ U . Hence we have(5)δl (a, y) = 0, for all y ∈ U.

Again replacing y by b in (3) and using (1), we get δl (x, b)σ([a, b]) = 0,
for every x ∈ U . Since σ([a, b]) is also a right nonzero divisor, we find that(6)δl (x, b) = 0, for all x ∈ U.

Combining (4), (5) and (6) we find that δl (x, y)σ([a, b]) = 0 and henceδl (x, y) = 0, for every x, y ∈ U , that is, G is a left σ-centralizer of U intoR.

Proof of Theorem 2.2. The proof will be done only for the left side case,because the right one is analogous.
By assumption we easily obtain(7)

G(xyz + zyx) = G(x)σ(y)σ(z) + G(z)σ(y)σ(x), ∀x, y, z ∈ U,

since R is 2-torsion-free and U is square closed.
Putting z = 2xy ∈ U in (7), we get G(xy · (2xy) + (2xy) · yx) =2G(x)σ(y)σ(x)σ(y) + 2G(xy)σ(y)σ(x), for every x, y ∈ U . On the other
hand, by assumption, we haveG(x(2yx)y) + 2G(xy 2 x) = 2G(x)σ(y)σ(x)σ(y) + 2G(x)σ(y 2 )σ(x),for every x, y ∈ U .
As before, we will denote δl (x, y) = G(xy) − G(x)σ(y), for every x, y ∈ U .
Thus, the expression above gives δl (x, y)σ(y)σ(x) = 0 for each x, y ∈ U .
Furthermore, by assumption, we haveG((2xy)xy) + G((2xy)yx) = 2G(xy)σ(x)σ(y) + 2G(xy)σ(y)σ(x), ∀ x, y ∈ U.
On the other hand,G(xy ·(2xy)+(2xy)·yx) = 2G(x)σ(y)σ(x)σ(y)+2G(xy)σ(y)σ(x), ∀x, y ∈ U.
Hence δl (x, y)xy = 0, for each x, y ∈ U , since R is 2-torsion-free.
Hence, we get δl (x, y) · σ([x, y]) = 0, for every x, y ∈ U .
Now we are ready following the same steps as in the proof of Theorem
2.1.
Proof of Theorem 2.3. By our assumptions on R, U and G: R → R, weeasily obtain(8)

2G(xyx) = G(x)σ(y)σ(x) + σ(x)σ(y)G(x),116

W. CORTES AND C. HAETINGER

for every x, y ∈ U , and2G(xyz + zyx) = G(x)σ(y)σ(z) + σ(x)σ(y)G(z)++G(z)σ(y)σ(x) + σ(z)σ(y)G(x),(9)

for each x, y, z ∈ U .
In particular, for z = 2xy ∈ U , we have2G(xy · (2xy)) + 4G(xy 2 x)

= 2G(x)σ(y)σ(x)σ(y) + σ(x)σ(y)G(2xy)++2G(x)σ(y 2 )σ(x) + 2σ(x)σ(y 2 )G(x),for every x, y ∈ U . On the other hand,
2G(xy · (2xy) + (2xy) · yx)= 2G(x)σ(y)σ(x)σ(y) + G(2xy)σ(y)σ(x)++2σ(x)σ(y 2 )G(x) + σ(x)σ(y)G(2xy),

for each x, y ∈ U .
Denoting again δl (x, y) = G(xy) − G(x)σ(y) for every x, y ∈ U as above,
we have that(10)δl (x, y) · σ(y)σ(x) = 0, ∀x, y ∈ U,

since R is 2-torsion-free.
Analogously, for z = 2yx ∈ U , we get
4G(xy 2 x) + 2G((2yx)yx)= 2G(x)σ(y 2 )σ(x) + 2σ(x)σ(y 2 )G(x)++G(2yx)σ(y)σ(x) + 2σ(y)σ(x)σ(y)G(x),for every x, y ∈ U . On the other hand,
2G(xy · (2yx) + (2yx) · yx)= 2G(x)σ(y 2 )σ(x) + G(2yx)σ(y)σ(x)++2σ(y)σ(x)σ(y)G(x) + σ(x)σ(y)G(2yx),

for each x, y ∈ U .
As above, denoting δr (x, y) = G(xy) − σ(x)G(y) for every x, y ∈ U , wehave that(11)

σ(x)σ(y) · δr (y, x) = 0, ∀x, y ∈ U,

since R is 2-torsion-free.
Let b ∈ U be a nonzero divisor. Swap now x by b in (10) to obtain(12)δl (b, y)σ(y) = 0, ∀y ∈ U,

since σ(b) is a nonzero divisor, too.
Replacing x by x + b in (10), and using (10) and (12), we getδl (x, y)σ(y)σ(b) = 0, for every x, y ∈ U . Since σ(b) is not a zero divisor,
we have that(13)

δl (x, y)σ(y) = 0, ∀x, y ∈ U.

Substitute y by b in (13) to obtain(14)

δl (x, b) = 0, ∀x ∈ U.

ON LIE IDEALS AND LEFT JORDAN σ-CENTRALIZERS

117

Changing y by y + b in (13), and using (13) and (14), we obtain
δl (x, y)σ(b) = 0, for every x, y ∈ U . Therefore δl (x, y) = 0, for all x, y ∈ U ,and G is a left σ-centralizer of U into R.
Replace x by b in (11) to obtain(15)

σ(y)δr (y, b) = 0, ∀y ∈ U.

Substituting x by x + b in (11), using (11) and (15), we obtain
σ(b)σ(y)δr (y, x) = 0, for each x, y ∈ U , and since σ(b) is a nonzero divisor, it follows that(16)

σ(y)δr (y, x) = 0, ∀x, y ∈ U.

Change now y by b in (16) to obtain(17)

δr (b, x) = 0, ∀x ∈ U.

Replacing y by y +b in (16), and using (16) and (17), we get σ(b)δr (y, x) =0, for every x, y ∈ U . Hence, δr (y, x) = 0, for all x, y ∈ U , because σ(b) is
a nonzero divisor, and G is a right σ-centralizer of U into R. The proof iscomplete.

The following example shows, for the sake of completeness, that the assumptions of our main Theorem 2.1 and [10, Proposition 1.4] are independent each other. This example is due to M. Bres̆ar who kindly allowed us
to include it here.
Example [4, Example]. A semiprime ring may not contain a commutator
nonzero divisor (after all, take commutative semiprime rings, or more generally, semiprime rings R containing a nonzero central idempotent elemente ∈ R such that eR is commutative). Conversely, a ring may contain a commutator nonzero divisor, but is not semiprime. For example, let R = T2 (A1 )be the ring of the 2 × 2 upper triangular matrices whose entries are elementsfrom the Weyl algebra A1 (polynomials in x, y such that xy − yx = 1). Then
R is not semiprime, but the commutator of scalar matrices generated by x
and y is the identity matrix.
Finally, we give some well-known examples of not semiprime rings thathave commutators nonzero divisors.
Example. Consider the 2 × 2 matrix algebra M2 (R) over any ring R with
1. Let Eij be the usual matrix units. Then the commutator [E12 , E21 ] =E11 − E22 is invertible.
When char(R) 6= 2, in M3 (R) we have that [E12 + E23 , E21 − E32 ] =E11 − 2E22 + E33 is a nonzero divisor.
Further examples can be seen in [4, Example].

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W. CORTES AND C. HAETINGER

Remark. In the year 1999, J. Vukman [8, Theorem 1] proved that if R is a2-torsion-free semiprime ring and G: R → R is an additive mapping suchthat 2G(x2 ) = G(x)x + xG(x) holds for every x ∈ R, then G is a left andright centralizer.
If an additive mapping G: R → R, where R is an arbitrary ring, is bothleft and right Jordan σ-centralizer, then obviously G satisfies the relation
2G(x2 ) = G(x)σ(x) + σ(x)G(x), for every x ∈ R. It seems natural to ask
whether the result of ([8], Theorem 1) is also true for the relation above on
Lie ideals. We are unable to answer this question.
Acknowledgments
The authors are grateful to M. Bres̆ar for his kindly contribution with
examples, and they are grately indebted to the referee for his (her) valuableadvice in the writing of this final version.
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Wagner Cortes
INSTITUTO DE MATEMÁTICA
UNIVERSIDADE FEDERAL DO RIO GRANDE DO SULPORTO ALEGRE-RS, 91509-900, BRAZIL
e-mail address: cortes@mat.ufrgs.br

ON LIE IDEALS AND LEFT JORDAN σ-CENTRALIZERS

Claus Haetinger
CENTRO DE CIÊNCIAS EXATAS E TECNOLÓGICASUNIVATESLAJEADO-RS, 95900-000, BRAZIL
e-mail address: chaet@univates.br
URL: http://ensino.univates.br/~chaet(Received February 23, 2007 )119