# Dependent Elements of Derivations on  Semiprime Rings

Authors: Faisal Ali; Muhammad Anwar Chaudhry
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We characterize dependent elements of a commuting derivationd on a semiprime ring R and investigate a decomposition of R using dependent elements of d. We show that there exist ideals U and V of R such that U⊕V is an essential ideal of R, U∩V={0}, d=0 on U, d(V)⊆V, and d acts freely on V.


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## Body

## 1. Introduction and Preliminaries

Murray and von   Neumann [1] and von   Neumann [2] introduced the notion of free action on abelian von Neumann algebras and used it for construction of
certain factors. Kallman [3] generalized the notion of free action of automorphisms
to von Neumann algebras, not necessarily abelian, by using implicitly the
dependent elements of an automorphism. Dependent elements of automorphisms were
later studied by Choda et al.  [4] in the context of C*-algebras.
Several other authors have studied dependent elements of automorphisms in the
context of operator algebras (see [5, 6] and references therein). A brief  account of dependent elements in W*-algebras has
also appeared in the book of Strătilă [7].It is well
known that allC* and von Neumann
algebras are semiprime rings; in particular a von Neumann algebra is prime if
and only if its centre consists of the scalar multiples of identity [8]. Thus a natural extension of
the notion of a dependent element of mappings on a C*-algebra or von
Neumann algebras is the study of this notion in the context of semiprime rings
and prime rings.Laradji and
Thaheem [9] initiated
the study of dependent elements of endomorphisms of semiprime rings and
generalized a number of results of [4] for semiprime rings. Recently, Vukman and Kosi-Ulbl
[10] and Vukman
[11, 12] have made further study of dependent elements of some mappings on prime
and semiprime rings.On one hand,
motivated by the work of Laradji and Thaheem [9], Vukman and Kosi-Ulbl
[10], and Vukman
[11, 12] on dependent elements of mappings of semiprime rings and on the other
hand by the work done by various researchers on commuting derivations on prime
and semiprime rings, we investigate some properties, not already investigated,
of dependent elements of commuting derivations on semiprime rings. We show that
the dependent elements of a commuting derivation of a semiprime ring are
central and form a commutative semiprime subring of R. We also show that for a commuting derivation d on a semiprime
ring R, there exist ideals U and V of R such that U⊕V is an essential
ideal of R, U∩V={0}, d=0 on U, d(V)⊆V, and zero is the only dependent element of d∖V, the restriction of d on V; that is, d acts freely on V.Throughout,R will represent
an associative ring with centre Z(R). The commutator xy−yx will be denoted
by [x,y]. We will use the basic commutator identities [xy,z]=[x,z]y+x[y,z] and [x,yz]=y[x,z]+[x,y]z. Recall that a ring R is semiprime if aRa=0 implies a=0 and is prime if aRb=0 implies a=0 or b=0. An additive mapping d:R→R is called a
derivation on R if d(xy)=d(x)y+xd(y) for all x,y∈R. It is called commuting if [d(x),x]=0 for all x∈R. Let a∈R, then the mapping d:R→R given by d(x)=[a,x] is a derivation
on R. It is called inner derivation on R.We call an
element elementa∈R a dependent
element of a derivation d:R→R if d(x)a=[a,x]a holds for all x∈R. Following [3], a derivation d:R→R is said to act
freely on R (or a free
action) in case zero is the only dependent element of d. It is known that a semiprime ring R has no central
nilpotent elements. We will use this fact without any specific reference. For
a derivation d:R→R, D(d) denotes the
collection of all dependent elements of d.It is known
that the left and right annihilators of an idealU of a semiprime
ring R coincide. It
will be denoted by Ann(U). It is also known that U∩Ann(U)={0} and U⊕Ann(U) is an essential
ideal of R. We will use these facts without any further
reference.We will use
the following result in the sequel.Theorem 1.1 (see [8,Corollary 3.2]).
Ifd is a commuting
inner derivation on a semiprime ring R, then d=0.


## 2. Results

We now prove our results.Theorem 2.1.
Letd be a commuting
derivation of a semiprimering R. Then a∈D(d) if and only if a∈Z(R) and d(x)a=0 for all x∈R.Proof.
Leta∈D(d). Then (2.1)d(x)a=[a,x]a∀x∈R. Replacing x by xy in (2.1), we
get d(xy)a=[a,xy]a. That is, (2.2)d(x)ya+xd(y)a=x[a,y]a+[a,x]ya∀x,y∈R. From (2.1) and (2.2), we get (2.3)d(x)ya=[a,x]ya∀x,y∈R. Multiplying (2.3) by z on the right,
we get (2.4)d(x)yaz=[a,x]yaz. Replacing y by yz in (2.3), we
get (2.5)d(x)yza=[a,x]yza. Subtracting (2.5) from (2.4), we get d(x)y(az−za)=[a,x]y(az−za), which implies (2.6)d(x)y[a,z]=[a,x]y[a,z]. Multiplying (2.6) by x on the left, we
get (2.7)xd(x)y[a,z]=x[a,x]y[a,z]. Replacing y by xy in (2.6), we
get (2.8)d(x)xy[a,z]=[a,x]xy[a,z]. Subtracting (2.7) from (2.8), we get (2.9)[d(x),x]y[a,z]=[[a,x],x]y[a,z]. Since d is commuting,
therefore from (2.9) we get (2.10)[[a,x],x]y[a,z]=0. From (2.10), we get (2.11)[[a,x],x]y[a,z]z=0. Replacing y by yz in (2.10) and
then subtracting the result from (2.11), we get [[a,x],x]y[[a,z],z]=0, which implies [[a,x],x]y[[a,x],x]=0. Using semiprimeness of R, from the last relation we get (2.12)[[a,x],x]=0∀x∈R. Thus inner derivation ψ:R→R defined by ψ(x)=[a,x] is commuting.
Hence ψ(x)=0 by Theorem 1.1,
which implies [a,x]=0. Thus a∈Z(R). Further from (2.1), we get d(x)a=0.
Conversely, leta∈Z(R) and let d(x)a=0. Then d(x)a=0=[a,x]a. So a∈D(d). This completes the proof.Corollary 2.2.
LetR be a semiprime
ring and let d be a commuting
derivation of R. Let a∈D(d), then d(a)=0.Proof.
Sincea∈D(d), therefore (2.13)d(x)a=0∀x∈R. Replacing x by d(x) in (2.13), we
get (2.14)d2(x)a=0∀x∈R. From (2.13), we get d(d(x)a)=d(0)=0, which implies d2(x)a+d(x)d(a)=0. Using (2.14), from the last relation we get (2.15)d(x)d(a)=0. Replacing x by ax in (2.15) and
using (2.15), we get 0=d(ax)d(a)=d(a)xd(a)+ad(x)d(a)=d(a)xd(a). Thus d(a)xd(a)=0 for all x∈R. Using semiprimeness of R, from the last equation we get d(a)=0.Corollary 2.3.
LetR be a semiprime
ring and let d be a commuting
derivation of R. Then D(d) is a
commutative semiprime  subring of R.Proof.
Leta,b∈D(d). Then by Theorem  2.1a,b∈Z(R), d(x)a=0, and d(x)b=0 for all x∈R. Obviously a−b∈Z(R) and d(x)ab=0. So a−b and ab∈D(d). Since a,b∈Z(R), so ab=ba. Thus D(d) is a
commutative subring of R. To show semiprimeness of D(d), we consider aD(d)a=0, a∈D(d). Then axa=0 for all x∈D(d). In particular a3=0, which implies a=0 because R has no central
nilpotents. Thus D(d) is a
commutative semiprime subring of ring.Corollary 2.4.
LetR be a
commutative semiprime ring and let d be a derivation
of R. Then D(d) is an ideal of R.Proof.
SinceR is commutative,
so d is commuting.
Let a,b∈D(d). Then by Corollary 2.3, a−b∈D(d). Let a∈D(d) and let r∈R. Then d(x)a=0 for all x∈R. Thus d(x)ar=0. Since ar=ra, so d(x)ar=d(x)ra=0 for all x∈R. Hence ar=ra∈D(d). Thus D(d) is an ideal of R.Remark 2.5.
(i)  IfR is a semiprime
ring and U an ideal of R, then it is easy to verify that U is a semiprime
subring of R and Z(U)⊆Z(R).
(ii)  Ifd is a commuting
derivation on R and a∈D(d), then by Theorem  2.1,
d(x)a=0 for all x∈R. This implies 0=d(xy)a=d(x)ya+xd(y)a=d(x)ya, which gives d(x)ya=0. Thus ad(x)yad(x)=0 for all x,y∈R, which by semiprimeness of R implies ad(x)=0.Theorem 2.6.
LetR be a semiprime
ring and let d be a commuting
derivation on R. Then there exist ideals U and V of R such that
(a)
U⊕V is an essential
ideal of R and U∩V={0},(b)
d=0 on U and d(V)⊆V,(c)
D(d∖V)={0}, where d∖V is restriction
of d on V. That is, d acts freely on V.Proof.
(a)  LetU be the ideal of R generated by D(d). Let V=Ann(U). Then V is an ideal of R, U⊕V is an essential
ideal of R, and U∩V={0}.
(b)  By Corollary2.2 and Theorem  2.1,
d(a)=0 and d(x)a=0 for all x∈R and a∈D(d). By Remark 2.5(ii) ad(x)=0. Thus d(ax)=d(a)x+ad(x)=0, d(xa)=d(x)a+xd(a)=0, and d(xay)=d(x)ay+xd(a)y+xad(y)=0 for a∈D(d) and x,y∈R. Hence d=0 on U.
Letv∈V=Ann(U). Thus va=0 for all a∈U. So d(va)=d(0)=0, which implies d(v)a+vd(a)=0. Thus d(v)a=0 because d=0 on U. So, d(v)∈Ann(U)=V. Hence d(V)⊆V.
(c)  SinceV is an ideal of R, so by Remark 2.5(i) V is a semiprime
subring of R and Z(V)⊆Z(R). Since d(V)⊆V, so d∖V is a derivation
on V. Let c∈V be a dependent
element of d∖V, so by Theorem  2.1  and Corollary 2.2  c∈Z(V)⊆Z(R), d∖V(c)=0, and d∖V(v)c=cd∖V(v)=0. Let x∈R, so xv∈V. Thus d∖V(xv)c=0, which implies d(xv)c=0. That is, d(x)vc+xd(v)c=0, which implies d(x)vc=0. Since V is an ideal of R, so cvd(x)∈V and d(x)c∈V for all x∈R. Replacing v by cvd(x) in d(x)vc=0, we get d(x)cvd(x)c=0. Using semiprimeness of V, we get d(x)c=0. Since c∈Z(R) and d(x)c=0 for all x∈R, therefore c∈D(d)⊆U. So c∈U and c∈V=Ann(U). Thus c=0. Hence D(d∖V)={0}. That is, d acts freely on V.Since every derivationd on a
commutative ring is a commuting derivation and D(d) is an ideal of R by Corollary
2.4, therefore in case of a commutative ring U=D(d) and V=Ann(D(d)). Thus we have the following corollary.Corollary 2.7.
LetR be a
commutative semiprime ring and let d be a derivation
on R. Then there exist ideals U=D(d) and V=Ann(D(d)) of R such that
(a)
U⊕V is an essential
ideal of R and U∩V={0},(b)
d=0 on U, d(V)⊆V,(c)
D(d∖V)={0}, where d∖V is restriction
of d on V. That is, d acts freely on V.The authors are
thankful to the referee for suggesting another proof of Theorem2.6 with
different ideals V and U. The ideal V is generated by d(R) and U=AnnR(V). The proof suggested by the referee is based on a
theorem of  Chuang and  Lee [13]. The statement of the said theorem and the proof of
Theorem 2.6 as suggested by the referee are given below.Theorem 2.8 (see [13]).
LetR be a semiprime
ring with a derivation d and let λ be a left ideal
of R. Suppose that [d(x),x]∈Z(R) for all x∈λ, where Z(R) denotes the
center of R. Then [λ,R]d(R)=0.Now we give the
proof of Theorem2.6 as suggested by the referee.Proof.
By
Theorem2.8, [R,R]d(R)=0. Let V be the ideal of R generated by d(R) and let U=AnnR(V). Clearly, [R,R]⊆U, U∩V=0, U⊕V is essential in R, and d(V)⊆V⊆Z(R). Since d(U)⊆U∩V, we have d(U)=0. Let c be a dependent
element of d∖V. Let v∈V. Then d(v)c=[c,v]c∈UV=0. Thus d(U⊕V)c=d(V)c=0. Since U⊕V is an essential
ideal of the semiprime ring R, we have d(R)c=0, implying that c∈U. So c∈U∩V=0, as asserted. This proves the theorem.



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*Source: 696737-2009-05-11.xml*