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Boletín de la Sociedad MatemáticaMexicanaThird Series ISSN 1405-213X Bol. Soc. Mat. Mex.DOI 10.1007/s40590-014-0029-3

On Lie ideals of $$*$$ ∗ -prime rings withgeneralized derivations

Nadeem Ur Rehman, RadwanMohammed AL-Omary & Abu ZaidAnsari

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Bol. Soc. Mat. Mex.DOI 10.1007/s40590-014-0029-3

ORIGINAL ARTICLE

On Lie ideals of ∗-prime rings with generalizedderivations

Nadeem Ur Rehman · Radwan Mohammed AL-Omary ·Abu Zaid Ansari

Received: 28 September 2013 / Accepted: 19 June 2014© Sociedad Matemática Mexicana 2014

Abstract Let (R, ∗) be a 2-torsion free ∗-prime ring with involution ∗, and L �= 0 bea ∗-Lie ideal of R. An additive mapping F : R → R is called a generalized derivationon R if there exists a derivation d: R → R such that F(xy) = F(x)y + xd(y) holdsfor all x, y ∈ R. In the present paper, we shall show that when L satisfies any ofseveral identities involving F , then L is central.

Keywords ∗-Lie ideal · ∗-Prime rings · Derivations · Generalized derivations

Mathematics Subject Classification (2000) 16W25 · 16N60 · 16U80

1 Introduction

Let R be an associative ring with center Z(R). A ring R is said to be n-torsion freeif nx = 0 implies x = 0 for all x ∈ R. For each x, y ∈ R, the symbol [x, y]will represent the commutator xy − yx and the symbol (x ◦ y) stands for the skew-commutator xy+yx . A ring R is prime if a Rb = {0} implies that either a = 0 or b = 0.

N. U. RehmanDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, Indiae-mail: rehman100@gmail.com

R. M. AL-OmaryDepartment of Mathematics, Faculty of Education, Al-Naderah, Ibb University, Ibb, Yemene-mail: radwan959@yahoo.com

A. Z. Ansari (B)Department of Mathematics, Faculty of Science, Islamic University in Madinah,Madinah, Kingdom of Saudi Arabiae-mail: ansari.abuzaid@gmail.com

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An additive mapping x �→ x∗ on a ring R is called an involution if (xy)∗ = y∗x∗ and(x∗)∗ = x hold for all x, y ∈ R. A left (resp. right, two sided) ideal I of R is called aleft (resp. right, two sided) ∗-ideal if I ∗ = I . An ideal P of R is called ∗-prime idealif P( �= R) is a ∗-ideal and for ∗-ideals I , J of R, I J ⊆ P implies that I ⊆ P or

J ⊆ P . An example: let Z be the ring of integers. Let R ={(

a b0 c

)| a, b, c ∈ Z

}.

We define a map ∗ : R −→ R as follows:

(a b0 c

)∗=

(c −b0 a

). It is easy to check that

I ={(

0 b0 0

)| b ∈ Z

}is a ∗−ideal of R. Now we give an example of ∗- prime ideal:

Let F be any field and R = F[x] be the polynomial ring over F . Let ∗ : R −→ Rbe a map defined by ∗( f (x)) = f (−x) for all f (x) ∈ R. Then it is easy to checkthat x R is a ∗-prime ideal of R. Note that an ideal I of R may be not a ∗-ideal: LetZ be the ring of integers and R = Z × Z. Consider a map ∗ : R −→ R definedby (a, b)∗ = (b, a) for all a, b ∈ R. For an ideal I = Z × {0} of R, I is not a∗-ideal of R since I ∗ = {0} × Z �= I . A ring R equipped with an involution ∗ issaid to be a ∗-prime ring if for any a, b ∈ R, a Rb = a Rb∗ = {0} implies a = 0 orb = 0. Obviously, every prime ring equipped with involution ∗ is ∗-prime. The set ofsymmetric and skew-symmetric elements of a ∗-ring will be denoted by S∗(R) i.e.,S∗(R) = {x ∈ R | x∗ = ±x}.

An additive subgroup L of R is said to be a Lie ideal of R if [L , R] ⊆ L . A Lieideal L is said to be non-commutative if [L , L] �= 0. Let L be a non-commutativeLie ideal of R. It is well known that [R[L , L]R, R] ⊆ L (see the proof of [6, Lemma1.3]). Since [L , L] �= 0, we have 0 �= [I, R] ⊆ L for I = R[L , L]R a nonzero idealof L . A Lie ideal is said to be a ∗-Lie ideal if L∗ = L . If L is a Lie (resp. ∗-Lie) idealof R, then L is called a square closed Lie (resp. ∗- Lie) ideal of R if x2 ∈ L for allx ∈ L .

An additive mapping d : R −→ R is called a derivation if d(xy) = d(x)y + xd(y)

for all x, y ∈ R. In particular, for fixed a ∈ R, the mapping Ia : R −→ R given byIa(x) = [a, x] is a derivation which is said to be an inner derivation.

An additive function F : R −→ R is called a generalized inner derivation ifF(x) = ax + xb for fixed a, b ∈ R. For such a mapping F , it is easy to see that

F(xy) = F(x)y + x[y, b] = F(x)y + x Ib(y) for all x, y ∈ R.

This observation leads to the following definition, an additive mapping F : R → Ris called a generalized derivation associated with a derivation d if F(xy) = F(x)y +xd(y) holds for all x, y ∈ R. It would be convenient to denote it by (F, d).

Familiar examples of generalized derivations are derivations and generalized innerderivations, and the latter includes left multipliers. Since the sum of two generalizedderivations is a generalized derivation, every map of the form F(x) = cx+d(x), wherec is a fixed element of R and d a derivation of R, is a generalized derivation; and if Rhas multiplicative identity 1, then all generalized derivations have this form. Over thelast four decades, several authors have proved commutativity theorems for prime ringsor semiprime rings admitting automorphisms, derivations or generalized derivationswhich are centralizing or commuting on appropriate subset of R (see [1–5,8–10]

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and [7], for partial bibliography). In this paper, we shall discuss when L ⊆ Z(R)

such that R is a ∗-prime ring with generalized derivations (F, d) and (G, g) satisfyingany one of the following properties: (i) F(x) = 0, (ii) [F(x), y] = (F(x) ◦ y), (iii)F(x2) = x2, (iv) F(x)x = xG(x), (v) [F(x), y] = [x, G(y)], (vi) F([x, y]) =[y, G(x)], (vii) (F(x) ◦ y) = (x ◦ G(y), for all x, y ∈ L .

2 Preliminary results

We shall use without explicit mention the following basic identities that hold for anyx, y, z ∈ R:

• [xy, z] = x[y, z] + [x, z]y• [x, yz] = y[x, z] + [x, y]zBefore we proceed, we state a few standard, well-known facts. For the sake of com-pleteness we sketch the proof of Facts B–E.Fact A In any 2-torsion free semiprime ring R, and for any Lie ideal L satisfying[L , L] = 0 must be central ( see the proof [6, Lemma 1.3]).Fact B In any 2-torsion free semiprime ring R, and for any non-central Lie ideal L ,the subring 〈L〉 generated by L contains the nonzero ideal I = R[L , L]R. Thus,when L is ∗-invariant then I is a nonzero ∗-ideal. In this case we will simply writeI ⊆ 〈L〉.Proof Assume that L is non-central Lie ideal. Clearly [L , L] ⊆ L2 for x, y ∈ R

r [x, y] = [r x, y] − [r, y]x ∈ L + L2.

But L2 is itself a Lie ideal, i.e., [xy, r ] = x[y, r ] + [x, r ]y; so, commuting R[x, y]with R we get R[L , L]R ⊆< L >. This ideal not zero since then R semiprime forcesL to be commutative and hence by Fact A, we get contradiction. When L is ∗-invariant,then R[L , L]R is a ∗-invariant ideal. This completes the proof of the Fact B. ��Fact C In any ∗-prime ring R, any nonzero ∗-ideal I has no left (right) annihilatorand [r, I ] = 0 implies that r ∈ Z(R).

Proof Let d be the inner derivation induced by r , that is d(x) = [r, x] for all x ∈ R.Then d(I ) = 0 because of [r, I ] = 0. Thus,

0 = d(I ) = d(I R) = d(I )R + I d(R) = I d(R) = I Rd(R).

Hence, I Rd(R) = 0 and also I ∗ Rd(R) = 0, since I = I ∗. As I �= 0, the ∗-primenessof R forces that d = 0. This implies that [r, s] = 0 for all s ∈ R, proving r ∈ Z(R).

��Fact D For R semiprime and L a Lie ideal, aLb = 0 implies that a〈L〉b = 0. Thus,for L a non-central ∗-Lie ideal is in a ∗-prime ring R with char R �= 2, aI b = 0.

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Proof If axb = 0 for all x ∈ L , then y ∈ L

0 = a[y, xbrayx]b = (ayxb)r(ayxb)

The semiprimeness of R forces that: aL2b = 0. By induction on power of L (let thex above be in Lm), aLmb = 0 for all m, so a(< L >)b = 0 and hence by abovea(R[L , L]R)b = 0. If L is ∗-invariant, the ideal R[L , L]R is ∗-invariant. ��Fact E Every ∗-prime ring is semiprime.

Proof Indeed, if a Ra = 0 then a Ra Ra∗ = 0 so that a = 0 or ara∗ = 0. Buta Ra∗ = 0 together with a Ra = 0, and hence we get a = 0. ��

3 ∗-Lie ideals and generalized derivations

Theorem 3.1 Let R be a ∗-prime ring with involution ∗ and L be a nonzero ∗-Lieideal of R. If R admits a generalized derivation (F, d) such that F(L) = 0 , thenF = d = 0 or L ⊆ Z(R).

Proof Assume that L is not a central Lie ideal. For x ∈ L and r ∈ R, [x, r ]x =[x, r x] ∈ L . Then 0 = F([x, r ]x) = F([x, r ])x + [x, r ]d(x), so we obtain that[x, r ]sd(x) = 0 for all x ∈ L , r, s ∈ R. Now, replacing x by x + y, we find that[x, r ]sd(y) + [y, r ]sd(x) = 0 for all x, y ∈ L and r, s ∈ R. Again, replacing s bysd(x)t in the above expression we find that [y, r ]sd(x)td(x) = 0; so we obtain

(d(x)s[y, r ]sd(x)) t (d(x)s[y, r ]sd(x)) = 0 for all x, y ∈ L and r, s, t ∈ R.

Thus, by Fact E and semiprimeness of R we get d(x)R[y, r ]Rd(x) = 0. Since L isnot central, I = R[L , R]L �= 0 is a ∗-ideal; so by Fact E and semiprimeness of R,

we obtain d(x) = 0 for all x ∈ L . Thus, d(L) = 0 implies that d(〈L〉) = 0, whichimplies that d(I ) = 0 and hence d(R) = 0. Now for x, y ∈ L , F(xy) = F(x)y = 0so F(L2) = 0, and from Fact C, F(L) = 0. By induction F(〈L〉) = 0; so F(I ) = 0,

giving F(R)I = F(RI ) = 0 and resulting in F(R) = 0. This completes the proof ofthe theorem. ��Theorem 3.2 Let R be a ∗-prime ring with char(R) �= 2 and L be a nonzero squareclosed ∗-Lie ideal of R. If R admits a nonzero generalized derivation (F, d) such that[F(x), y] = (F(x) ◦ y) for all x, y ∈ L, then either d = 0 or L ⊆ Z(R).

Proof Assume that L is not central. We have

[F(x), y] = (F(x) ◦ y) for all x, y ∈ L . (3.1)

This can be rewritten as F(x)y − yF(x) = F(x)y + yF(x) for all x, y ∈ L . Sincechar(R) �= 2, we get yF(x) = 0; so 0 = L F(〈L〉) = L F(I ). Since L is not central,I F(I ) = 0 and so F(I ) = 0. Thus, by Theorem 3.1, we obtain d = 0. ��

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Theorem 3.3 Let R be a ∗-prime ring with char(R) �= 2 and L be a nonzero squareclosed ∗-Lie ideal of R. If R admits a nonzero generalized derivation (F, d) such thatF(x2) = x2 for all x ∈ L, then either d = 0 or L ⊆ Z(R).

Proof Assume that L is not central. Given that

F(x2) = x2 for all x ∈ L .

Replacing x by x + y in the above relation, we get

F(x2 + y2 + xy + yx) = x2 + y2 + xy + yx for all x, y ∈ L . (3.2)

Using the given hypothesis in (3.2), we obtain F(xy+yx) = xy+yx, for all x, y ∈ L .

This can be written as F(x ◦ y) − (x ◦ y) = 0 for all x, y ∈ L . Replace y by 2yx anduse the last expression and the fact that char(R) �= 2 to get (x ◦ y)d(x) = 0. Again,replacing y by 2yz, we get

[x, y]zd(x) = 0 for all x, y, z ∈ L; (3.3)

i.e., [x, y]Ld(x) = 0 for all x, y ∈ L . Since L is not central, [x, y]I d(x) = 0and hence by Fact D either [x, y] = 0 or d(x) = 0. Let U = {x ∈ L | [x, y] =0 for all y ∈ L} and V = {x ∈ L | d(x) = 0}. Thus, U and V are additive subgroupsof L and U ∪ V = L . But (L ,+) is not a union of proper subgroups. Therefore, wehave either U = L or V = L . If U = L , then [x, y] = 0 for all x, y ∈ L and henceby Fact A we get a contradiction. In the second case, if V = L , then d(L) = 0 andhence d(〈L〉) = 0 implies that d(I ) = 0 �⇒ d(R) = 0. ��

Theorem 3.4 Let R be a 2-torsion free ∗-prime ring and L be a nonzero square closed∗-Lie ideal of R. Suppose R admits a pair of generalized derivations (F, d) and (G, g)

such that

(i) F(x)x = xG(x) for all x ∈ L or(ii) F([x, y]) = [y, G(x)] for all x, y ∈ L or

(iii) [F(x), y] = [x, G(y)] for all x, y ∈ L or(iv) (F(x) ◦ y) = (x ◦ G(y) for all x, y ∈ L .

If d �= 0 and g �= 0, then L ⊆ Z(R).

Proof (i) By hypothesis, we have

F(x)x = xG(x) for all x ∈ L .

On linearizing the above relation we find that

F(x)y + F(y)x = xG(y) + yG(x) for all x, y ∈ L . (3.4)

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We replace x by 2xy in (3.4) and use the fact that char(R) �= 2 to get

F(x)y2 + xd(y)y + F(y)xy = xyG(y) + yG(x)y + yxg(y) for all x, y ∈ L .

(3.5)

Right multiplication by y to the relation (3.4) yields that

F(x)y2 + F(y)xy = xG(y)y + yG(x)y for all x, y ∈ L . (3.6)

Combining (3.5) and (3.6), we obtain

xd(y)y = yxg(y) + x[y, G(y)] for all x, y ∈ L . (3.7)

Now, replacing x by 2zx in (3.7) and using the fact that char(R) �= 2, we get

zxd(y)y = yzxg(y) + zx[y, G(y)] for all x, y, z ∈ L . (3.8)

Left multiplying (3.7) by z, we arrive at

zxd(y)y = zyxg(y) + zx[y, G(y)] for all x, y, z ∈ L . (3.9)

From (3.8) and (3.9), we get [y, z]xg(y) = 0. This is same as (3.3) and hence usinga similar arguments as used in the last paragraph of Theorem 3.3, we get the requiredresult.

(ii) It is given that F and G are generalized derivations of R such that F([x, y]) =[y, G(x)]. If F = 0, then [y, G(x)] = 0 for all x, y ∈ L . Replacing x by 2xy andusing the fact that char(R) �= 2, we get x[y, g(y)] + [y, x]g(y) = 0 for all x, y ∈ L .Again, replacing x by 2zx and using the fact that char(R) �= 2, we get [y, z]Lg(y) = 0for all x, y, z ∈ L . Now using similar arguments as above we get the required result.On the other hand, if G = 0, then we have F([x, y]) = 0 for all x, y ∈ L , and henceby Theorem of [7] we get [y, z]Lg(y) = (0) for all x, y, z ∈ L . Therefore, proceedingin the same way as above we obtain L ⊆ Z(R). Henceforth, we shall assume thatd �= 0. For any x, y ∈ L we have

F([x, y]) = [y, G(x)]. (3.10)

Replacing y by 2yx in (3.10) and using the fact that char(R) �= 2, we get [x, y]d(x) =y[x, G(x)] for all x, y ∈ L . Again, replacing y by 2zy to get [x, z]yd(x) = 0, that is[x, z]Ld(x) = 0. Now using similar arguments as above in the last paragraph of theproof of Theorem 3.3, we get the required result.

(iii) It is given that F and G are generalized derivations of R such that

[F(x), y] = [x, G(y)] for all x, y ∈ L .

If F = 0, then we have [x, G(y)] = 0 for all x, y ∈ L . Using in the same manneras above we obtain that L ⊆ Z(R). On the other hand if G = 0, then we have

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[F(x), y] = 0 for all x, y ∈ L; replacing x by 2xy in the last expression and usingthe fact that char(R) �= 2, we get x[d(y), y] + [x, y]d(y) = 0. Again replacing xby 2zx using the last expression and char(R) �= 2, we get [z, y]xd(y) = 0. Now,applying similar techniques as used after (3.3) in the proof of Theorem 3.3 (i) yieldsthe required result. Henceforth, we shall assume that g �= 0, then we have

[F(x), y] = [x, G(y)] for allx, y ∈ L .

Replacing y by 2yx in the above expression and using the fact that char(R) �= 2, weobtain

y[F(x), x] = [x, y]g(x) + y[x, g(x)] for all x, y ∈ L . (3.11)

Again, replace y by 2zy in (3.11) to get [x, z]yg(x) = 0 for all x, y, z ∈ L . The lastexpression as the (3.3) and the result follows.

(iv) It is given that F and G are generalized derivations of R such that (F(x)◦ y) =(x ◦ G(y)). If F = 0, then we have x ◦ G(y) = 0 for all x, y ∈ L . Replacing y by2yx and using the fact that char(R) �= 2, we get (x ◦ y)g(x) − y[x, g(x)] = 0 forall x, y ∈ L . Again replacing y by 2zy in the last expression and using the fact thatchar(R) �= 2, we get [x, z]yg(x) = 0 for all x, y, z ∈ L . Now, applying a similartechnique as used in the above expression, we get that L ⊆ Z(R). On the other handif G = 0, then we have F(x) ◦ y = 0. Replacing y by 2yx in the last expression andusing the fact that char(R) �= 2, we get y[F(x), x] = 0 i.e., L[F(x), x] = 0 for allx ∈ L and hence [F(x), x] = 0 for all x ∈ L . Then [F(x), y] + [F(y), x] = 0 forall x, y ∈ L . Now, replacing y by 2yx and using the fact that R is 2-torsion free, wefind that y[d(x), x] + [y, x]d(x) = 0. Again replace y by 2zy to get [z, x]yd(x) = 0for all x, y, z ∈ L . Notice that the arguments given in the proof of theorem 3.3 after(3.3) are still valid in the present situation and hence repeating the same process, weget the required result. Henceforth, we shall assume that d �= 0 (or g �= 0), for anyx, y ∈ L we have

(F(x) ◦ y) = (x ◦ G(y)).

Replacing y by 2yx in the above expression and using the fact that char(R) �= 2, weget −y[F(x), x] = (x ◦ y)g(x) − y[x, g(x)] for all x, y ∈ L . Again replacing y by2zy and using the fact that char(R) �= 2, we get [x, z]yg(x) = 0 for all x, y, z ∈ L .Now using a similar argument as above, we get the required result.

Example 3.1 Let Z be a ring of integers and consider R ={(

a b0 0

)| a, b ∈ Z

}and

L ={(

0 b0 0

)| b ∈ Z

}. Define ∗ : R → R by

(a b0 0

)∗=

(0 −b0 a

), d : R → R by

d

(a b0 0

)=

(0 −b0 0

), F : R → R by F

(a b0 0

)=

(a 00 0

). Then R is a ring under

the usual operations with involution ∗, L is a square closed ∗-Lie ideal and it is easyto see that (F, d) is a generalized derivation of R satisfying the following properties:

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(i) F(x) = 0, (ii) [F(x), y] = (F(x) ◦ y), (iii) F(x2) = x2 for all x, y ∈ L , but L isnot central. Hence, in Theorem 3.1, Theorem 3.2 and Theorem 3.3, the hypothesis ofprimeness cannot be omitted.

Example 3.2 Consider the ring R, involution ∗, ∗-Lie ideal L and generalized deriva-tion (F, d) as in Example 3.1. Define mappings g, G : R → R such that g(x) =e11x −xe11 and G(x) = 2e11x −xe11 for all x ∈ R. Then it is easy to see that (G, g) isa generalized derivation of R satisfying the following properties: (i) F(x)x = xG(x),(ii) [F(x), y] = [x, G(y)], (iii) F([x, y]) = [y, G(x)], (iv) (F(x) ◦ y) = (x ◦ G(y))

and for all x, y ∈ L , but L is not central. Hence in Theorem 3.4, the hypothesis ofprimeness cannot be omitted.

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