Generalized skew derivations implemented by elementary operators

GENERALIZED SKEW DERIVATIONS IMPLEMENTED BY ELEMENTARY OPERATORS ´ AND DIJANA ILISEVI ˇ ´ DANIEL EREMITA, ILJA GOGIC, C

Abstract. Let R be a semiprime ring with the maximal right ring of quotients Qmr . An additive map d : R → Qmr is called a generalized skew derivation if there exists a ring endomorphism σ : R → R and a map δ : R → Qmr such that d(xy) = δ(x)y + σ(x)d(y) for all x, y ∈ R. If σ is surjective, we determine the structure of generalized skew derivations for which there exists a finite number of elements ai , bi ∈ Qmr such that d(x) = a1 xb1 + · · · + an xbn for all x ∈ R.

1. Introduction By the main result of Pedersen in [8], every derivation of a separable C ∗ -algebra A is implemented as an inner derivation by an element of the local multiplier algebra Mloc (A), where Mloc (A) is the C ∗ -direct limit of the multiplier algebras of the closed two-sided essential ideals of A (see [1] for details). However, this problem is still wide open in the inseparable case. Note that derivations of C ∗ -algebras share some nice properties with the operators lying in the completely bounded norm closure E(A)cb of the∑ set E(A) of all elementary operators on A (i.e. maps ψ : A → A such n that ψ(x) = i=1 ai xbi for some elements ai , bi in the multiplier algebra M (A) of A): for example, every derivation of a C ∗ -algebra A is completely bounded, it is linear over the center of M (A), and it preserves the closed two-sided ideals of A. Motivated by this observation, the second author in [5, 6] considered the problem of characterizing the class of C ∗ -algebras A which admit only inner derivations inside E(A)cb . In particular, we showed that the answer to this problem heavily depends on the ideal structure of A (see [6, Theorem 1.5] and [5, Example 6.1]). However, by [6, Corollary 3.2] every derivation of A which lies in E(A)cb is always implemented by a local multiplier (i.e. by an element of Mloc (A)). The purpose of this paper is to obtain a similar result for semiprime rings R. In this purely algebraic setting, we replace the local multiplier algebra Mloc (A) by the maximal right ring of quotients Qmr of R, and we replace derivations by a larger class of additive maps ∑n d : R → Qmr which are implemented by elementary operators, that is d(x) = i=1 ai xbi for some ai , bi ∈ Qmr . More precisely, we shall study the following class of maps (see also [7]): 2010 Mathematics Subject Classification. Primary 16W20, 16W25. Secondary 16N60, 47B47. Key words and phrases. (generalized) skew derivation, elementary operator, epimorphism, semiprime ring, maximal (right) ring of quotients. The third author was supported by the Ministry of Science, Educations and Sports of the Republic of Croatia (Grant 037-0372784-2757). 1

´ AND DIJANA ILISEVI ˇ ´ DANIEL EREMITA, ILJA GOGIC, C

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Definition 1.1. Let σ : R → R be a ring endomorphism. An additive map d : R → Qmr is called a generalized σ-derivation (or a generalized skew derivation) if there exists a map δ : R → Qmr such that (1.1)

d(xy) = δ(x)y + σ(x)d(y)

for all x, y ∈ R. If there exist p, q ∈ Qmr such that d(x) = px + σ(x)q for all x ∈ R, then we say that d is an inner generalized σ-derivation. In this case we also say that the pair (p, q) implements d. Note that the map δ is automatically additive and it is uniquely determined by d. The main advantage of generalized σ-derivations is the fact that they simultaneously generalize σ-derivations from R to Qmr (we get them for δ = d, see Section 4) and left R-module σ-homomorphisms from R to Qmr (we get them for δ = 0, see Definition 3.1). In fact every generalized σ-derivation is a sum of these two type of maps (Remark 4.2). We now state the main result of this paper: Theorem 1.2. Let R be a semiprime ring and let d : R → Qmr be a generalized σ-derivation, where σ is surjective. Then the following conditions are equivalent: (i) d is implemented by an elementary operator; (ii) d is inner, and for each pair (p, q) which implements d there exists an invertible element a ∈ Qmr such that ε(q)σ(x) = ε(q)axa−1 for all x ∈ R; hence d(x) = px + axa−1 q

(x ∈ R).

By ε(q) we denote the central support of q (see Section 2 for details). Theorem 1.2 will be proved in Section 4. 2. Preliminaries Throughout this paper, R will be a semiprime ring. One can form the maximal right ring of quotients Qmr = Qmr (R) of R (see e.g. [2, Section 2.1]). This concept was introduced by Utumi [9], therefore the ring Qmr is also called Utumi right ring of quotients. The basic and in fact the characteristic four properties of Qmr are: (i) R is a subring of Qmr , (ii) for any q ∈ Qmr there exists a dense right ideal I of R such that qI ⊆ R, (iii) if 0 ̸= q ∈ Qmr and I is a dense right ideal of R, then qI ̸= 0, (iv) for any dense right ideal I of R and a right R-module homomorphism f : I → R there exists q ∈ Qmr such that f is a left multiplication by q. The center of Qmr is called the extended centroid of R and it is denoted by C = C(R). It is well known that C is a von Neumann regular self-injective (unital commutative) ring (see e.g. [2, Theorem 2.3.9 (iii)]). We shall mostly deal with central idempotents. The set of all idempotents in C will be denoted by Idem(C). Given ε, ε′ ∈ Idem(C), we write ε′ ≤ ε if ε′ ε = ε′ .

GENERALIZED SKEW DERIVATIONS IMPLEMENTED BY ELEMENTARY OPERATORS 3

For any subset S ⊆ Qmr there exists a unique ε(S) ∈ Idem(C) such that rannQmr (Qmr SQmr ) = (1 − ε(S))Qmr

and ε(S)x = x for all x ∈ S,

where rannQmr (X) denotes the right annihilator in Qmr of a subset X ⊆ Qmr (see e.g. [2, Theorem 2.3.9 (i)]). The idempotent ε(S) is called the central support of S. Whenever S = {x} for some x ∈ Qmr we write ε(x) for ε(S). Although the next two facts seem to be well known, we include their proofs for the convenience of the reader. Proposition 2.1. For all x ∈ Qmr and c ∈ C we have ε(cx) = ε(c)ε(x). Proof. Obviously ε(cx) ≤ ε(c)ε(x). Conversely, since C is von Neumann regular, there exists a unique element c′ ∈ C such that c′ c = ε(c) = ε(c′ )

(2.1)

(see [1, Corollary 2.1.8]). Hence, using [2, Theorem 2.3.9 (ii)] we have ε(cx) ≥ ε(c′ cx) = ε(ε(c)x) = ε(c)ε(x),

which shows ε(cx) = ε(c)ε(x). Proposition 2.2. For all ε ∈ Idem(C) we have Qmr (εR) = εQmr (R).

Proof. Let S := εR ⊕ (1 − ε)R. Since R ⊆ S ⊆ Qmr (R), [2, Proposition 2.1.10] implies Qmr (S) = Qmr (R). Moreover, since εR is an ideal of a semiprime ring S and rannS (εR) = (1 − ε)R, [2, Corollary 2.1.12] yields Qmr (R) = Qmr (S) = Qmr (εR) ⊕ Qmr ((1 − ε)R). Using [2, Theorem 2.3.9 (ii)] it is easy to see that ε is the identity element in Qmr (εR) and 1 − ε is the identity element in Qmr ((1 − ε)R). Thus, it follows that Qmr (εR) = εQmr (R). We shall also frequently use the following well known facts. Proposition 2.3 (Lemma 2.3.10 in [2]). Let S, T be arbitrary subsets of Qmr . Then the following conditions are equivalent: (i) (ii) (iii) (iv)

SIT = 0 for some dense right ideal I of R; T Qmr S = 0; ε(S)T = 0; ε(T )ε(S) = 0.

Theorem 2.4 (Theorem 2.3.9 (iv) in [2]). Any finitely generated submodule M = ∑n i=1 Cbi of the C-module Qmr contains a finite subset of nonzero elements {b′1 , . . . , b′k }, where k is the minimal number of generators of M , such that M=

k ⊕ i=1

Cb′i

and

ε(b′i+1 ) ≤ ε(b′i ) for all 1 ≤ i ≤ k − 1.


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Definition 2.5. We say that a map ψ : R → Qmr is implemented by an elementary operator (or simply, ψ is an elementary operator ) if ψ can be expressed as a finite sum n ∑ ψ= Mai ,bi i=1

of two-sided multiplication operators Mai ,bi : x 7→ ai xbi

(x ∈ R),

∑n where the coefficients ai , bi are elements of Qmr . In this case we say that i=1 Mai ,bi is a representation of ψ of length n. If ψ ̸= 0, the length ℓ(ψ) of ψ is defined as the least positive integer that arises as a length of a representation of ψ. We also define ℓ(0) := 0. Next we state the following version of a result which is due to Erickson, Martindale, and Osborn [4, Theorem 3.1]. ∑n Theorem 2.6 (Theorem 2.3.3 in [2]). Let q1 , q2 , . . . , qn ∈ Qmr . If q1 ̸∈ i=2 Cqi then there exists an elementary operator ψ : R → R with all coefficients lying in R such that ψ(q1 ) ̸= 0,

but

ψ(qi ) = 0 for all 2 ≤ i ≤ n.

The next result is a generalization of [2, Lemma 6.3.12]. Lemma 2.7. Let I be a dense right ideal of R and let b1 , . . . , bn ∈ Qmr . For a given set S and functions f1 , . . . , fn : S → Qmr we define a map ϕ : S × I → Qmr by ϕ(s, x) :=

n ∑

fi (s)xbi .

i=1

If ϕ(s, x) = 0 for all s ∈ S and x ∈ I, then ε(fi (S))bi ∈

∑

Cbj for all 1 ≤ i ≤ n.

j̸=i

In particular, if

∑n i=1

Cbi =

⊕n i=1

Cbi , then the following conditions are equivalent:

(i) ϕ(s, x) = 0 for all s ∈ S and x ∈ I; (ii) ε(fi (S))bi = 0 for all 1 ≤ i ≤ n. ∑ Proof. Let εi := ε(fi (S)). On the contrary, suppose that εi bi ̸∈ j̸=i Cbj for some ∑n 1 ≤ i ≤ n. Without loss of generality we can assume b′1 := ε1 b1 ̸∈ i=2 Cbi . ′ Let bi := bi for 2∑≤ i ≤ n. By Theorem 2.6 there exists an elementary operator m ψ : R → R, ψ = j=1 Muj ,vj (uj , vj ∈ R) such that (2.2)

ψ(b′1 ) =

m ∑ j=1

uj b′1 vj ̸= 0,

but ψ(b′i ) =

m ∑ j=1

uj b′i vj = 0 for all 2 ≤ i ≤ n.


Note that ε1 ψ(b′1 ) = ψ(b′1 ) ̸= 0. Using (2.2) we get f1 (s)xψ(b′1 ) = =

=

n ∑

fi (s)xψ(b′i ) =

i=1 m ∑ n ∑

n ∑ m ∑

fi (s)xuj b′i vj

i=1 j=1 m ∑ n ∑

fi (s)xuj b′i vj =

j=1 i=1 m ∑

fi (s)xuj bi vj

j=1 i=1

ϕ(s, xuj )vj = 0

j=1

for all s ∈ S and x ∈ I. Then Proposition 2.3 implies ε1 ψ(b′1 ) = 0, a contradiction. 3. Left R-module σ-homomorphisms implemented by elementary operators The goal of this section is to determine the structure of left R-module σ-homomorphisms ρ : R → Qmr , where σ : R → R is a ring epimorphism, which are implemented by elementary operators (Theorem 3.3). This result will be also used in the proof of Theorem 1.2. Definition 3.1. Let σ : R → R be a ring endomorphism. We say that an additive map ρ : R → Qmr is a left R-module σ-homomorphism (or a left R-module skew homomorphism) if ρ(xy) = σ(x)ρ(y) for all x, y ∈ R. Example 3.2. Each element q ∈ Qmr induces a left R-module σ-homomorphism ρ : R → Qmr given by ρ(x) := σ(x)q. We now state the main result of this section. Theorem 3.3. Let ρ : R → Qmr be a left R-module σ-homomorphism, where σ is surjective. Then the following conditions are equivalent: (i) ρ is implemented by an elementary operator; (ii) there exists an element q ∈ Qmr , and an invertible element a ∈ Qmr such that ρ(x) = σ(x)q and ε(q)σ(x) = ε(q)axa−1 for all x ∈ R; hence ρ(x) = axa−1 q

(x ∈ R).

In the sequel, for a surjective (not necessarily additive or multiplicative) function σ : R → R we define the set Mσ := {a ∈ Qmr : ax = σ(x)a for all x ∈ R}. Obviously, Mσ is a C-submodule of a C-module Qmr . In order to prove Theorem 3.3, we shall need the following generalization of [2, Proposition 3.1.12]. Lemma 3.4. Let σ : R → R be a surjective function.

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(i) Let a ∈ Mσ . If x ∈ Qmr is an element such that ax = 0 or xa = 0, then ε(a)x = 0. (ii) If a, a′ ∈ Mσ are elements such that ε(a′ ) ≤ ε(a), then a′ ∈ Ca. (iii) Mσ is a cyclic C-module. (iv) If σ is a ring epimorphism, then each element a ∈ Mσ is invertible in ε(a)Qmr . Before proving Lemma 3.4, let us recall that a subset S ⊆ Qmr is said to be orthogonally complete if for any dense orthogonal subset U ⊆ Idem(C) (i.e. ε(U ) = 1 and uv = 0 for all u ̸= v ∈ U ) and any subset {su : u ∈ U } ⊆ S there exists an element s ∈ S such that su = su u for all u ∈ U . The element s ∈ S is uniquely determined by the conditions su = su u for all u ∈ U , and it is denoted by s=

⊥ ∑

su u.

u∈U

More details can be found in [2, Section 3.1]. Proof of Lemma 3.4. (i). If ax = 0 then aRx = σ(R)ax = 0. Hence, ε(a)x = 0 by Proposition 2.3 (note that the assumption of surjectivity of σ is superfluous in this part). Similarly, if xa = 0 then xRa = xσ(R)a = xaR = 0, and again by Proposition 2.3 we conclude that ε(a)x = 0. (ii). On the contrary, suppose that∑a′ ∈ / Ca. By Theorem 2.6, there exists an m elementary operator ψ : R → R, ψ = j=1 Muj ,vj (uj , vj ∈ R) such that ψ(a′ ) =

m ∑

uj a′ vj ̸= 0,

but

ψ(a) =

j=1

m ∑

uj avj = 0.

j=1

Since a, a′ ∈ Mσ , this is equivalent to xa′ ̸= 0,

∑m

but xa = 0,

where x := j=1 uj σ(vj ). By the part (i) we conclude that ε(a)x = 0. Since ε(a′ ) ≤ ε(a), we obtain xa′ = xε(a)a′ = ε(a)xa′ = 0, a contradiction. (iii). We first show that Mσ is an orthogonally complete subset of Qmr . Indeed, let U be a dense orthogonal subset of Idem(C) and su ∈ Mσ for u ∈ U . Since Qmr ∑⊥ itself is orthogonally complete [2, Proposition 3.1.10], the element s := u∈U su u is a well defined element of Qmr . We claim that s ∈ Mσ . Indeed, by [2, Remark 3.1.9] for x ∈ R we have sx =

⊥ ∑ u∈U

su xu =

⊥ ∑

σ(x)su u = σ(x)s.

u∈U

By [2, Proposition 3.1.11] there exists an element a ∈ Mσ such that ε(a) = ε(Mσ ). In particular, this implies ε(a′ ) ≤ ε(a) for all a′ ∈ Mσ . Thus, by part (ii) we conclude that Mσ = Ca.


(iv). First assume that σ is an automorphism. By [2, Proposition 3.1.12] there exists u ∈ Mσ such that Mσ = Cu. Moreover, by [2, Proposition 3.1.12] u is invertible in ε(u)Qmr , and let u′ ∈ ε(u)Qmr be such that uu′ = u′ u = ε(u). Since a ∈ Mσ = Cu, we have a = cu for some c ∈ C. By [1, Corollary 2.1.8] there exists a unique element c′ ∈ C satisfying (2.1). By Proposition 2.1 we have ε(c)ε(u) = ε(cu) = ε(a), so c′ u′ ∈ ε(a)Qmr , and a(c′ u′ ) = cuc′ u′ = ε(c)ε(u) = ε(cu) = ε(a). Similarly, (c′ u′ )a = ε(a). This shows that a is invertible in ε(a)Qmr when σ is an automorphism. More generally, when σ is an epimorphism, we define a map τ : ε(a)R → ε(a)R by τ (ε(a)x) := ε(a)σ(x) (x ∈ R). Note that τ is a well-defined automorphism of ε(a)R. Indeed, a ∈ Mσ and σ(R) = R, together with Proposition 2.3 and Lemma 3.4 (i) imply that for all x ∈ R we have ε(a)x = 0 ⇔

axy = 0 for all y ∈ R

⇔ ⇔

σ(x)σ(y)a = 0 for all y ∈ R σ(x)Ra = 0

⇔

ε(a)σ(x) = 0.

Using Proposition 2.2 we have a ∈ ε(a)Qmr = Qmr (ε(a)R), so a lies in Mτ . It remains to apply the first part of the proof to conclude that a is invertible in ε(a)Qmr (ε(a)R) = ε(a)Qmr . Proof of Theorem 3.3. (ii) ⇒ (i). This is trivial. (i) ⇒ (ii). We assume ∑n that ρ ̸= 0, and suppose that ρ is implemented by an elementary operator i=1 Mai ,bi . By Theorem 2.4 there exist elements b′1 , . . . , b′k ∈ ⊕k ∑n ∑n ′ i=1 Cbi . Then i=1 Cbi = i=1 Cbi such that n ∑ i=1

Mai ,bi =

k ∑

Ma′j ,b′j ,

j=1

∑n for some elements a′j ∈ i=1 Cai . Hence, without loss of generality we can assume ∑n ⊕n that i=1 Cbi = i=1 Cbi . Since ρ satisfies ρ(xy) = σ(x)ρ(y), we have n ∑ (ai x − σ(x)ai )ybi = 0 i=1

for all x, y ∈ R. Using Lemma 2.7 and Proposition 2.3, we conclude that ε(bi )ai x = σ(x)ε(bi )ai for all x ∈ R and 1 ≤ i ≤ n. This means that each a′i := ε(bi )ai lies in Mσ , so in particular Mσ ̸= {0}. By Lemma 3.4 (iii) there exists an element u ∈ Mσ such that Mσ = Cu, and choose c1 , . . . , cn ∈ C such that a′i = ci u for each 1 ≤ i ≤ n.

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Now, for all x ∈ R, we have ρ(x)

n ∑

=

ai xbi =

i=1 n ∑

=

n ∑

a′i xbi

i=1

(

ci uxbi = ux

i=1

n ∑

) ci bi

= uxv

i=1

= σ(x)q, where v :=

n ∑

ci bi

and

q := uv.

i=1

By Lemma 3.4 (iv), u is invertible in ε(u)Qmr . Then the element a := u + 1 − ε(u) ∈ Qmr is invertible in Qmr (with inverse a−1 = u′ + 1 − ε(u), where u′ is the inverse of u in ε(u)Qmr ). Since u = ε(u)a, we have ε(u)v = a−1 q. Thus σ(x)q

=

uxv = ux(ε(u)v)

= (ε(u)a)x(a−1 q) = ax(ε(u)a−1 q) = axa−1 q for all x ∈ R. Consequently, axya−1 q

= σ(xy)q = σ(x)σ(y)q = σ(x)aya−1 q

for all x, y ∈ R, which implies (ax − σ(x)a)Ra−1 q = 0 for all x ∈ R. By Proposition 2.3, this is equivalent to ε(q)σ(x) = ε(q)axa−1 , since ε(bq) = ε(qb) = ε(q) for each invertible element b ∈ Qmr . This completes the proof. The following results are direct consequences of Theorem 3.3. First recall that an automorphism σ : R → R is said to be X-inner if there exists an invertible element a ∈ Qmr such that σ(x) = axa−1

(x ∈ R).

Corollary 3.5. A ring epimorphism σ : R → R is implemented by an elementary operator if and only if σ is an X-inner automorphism of R. Corollary 3.6. If R is a prime ring, and σ : R → R is a ring epimorphism, then σ is an X-inner automorphism of R if and only if there exists a non-zero left R-module σ-homomorphism ρ : R → Qmr which is implemented by an elementary operator.


4. Generalized σ-derivations implemented by elementary operators As already mentioned in Introduction, in this section we prove Theorem 1.2. Recall, if σ : R → R is a ring endomorphism, then a map δ : R → Qmr is said to be a σ-derivation (skew derivation) if δ satisfies the identity (4.1)

δ(xy) = δ(x)y + σ(x)δ(y)

for all x, y ∈ R. If there exists an element p ∈ Qmr such that δ(x) = px − σ(x)p for all x ∈ R, then we say that δ is an inner σ-derivation (and we also say that p implements δ). Remark 4.1. Note that σ-derivations include (standard) derivations (we get them for σ = idR ). Analogously, generalized σ-derivations include generalized derivations (see [3]). Remark 4.2. Each generalized σ-derivation d : R → Qmr can be decomposed as d = δ +ρ, where δ : R → Qmr is a σ-derivation, and ρ : R → Qmr is a left R-module σ-homomorphism. Indeed if x, y, z ∈ R, then (1.1) implies d(xyz) = δ(xy)z + σ(xy)d(z) = δ(xy)z + σ(x)σ(y)d(z), and d(xyz) = δ(x)yz + σ(x)d(yz) = δ(x)yz + σ(x)δ(y)z + σ(x)σ(y)d(z). Subtracting these two equations, one obtains (4.1) for all x, y ∈ R. Hence, δ is a σ-derivation. It is now easy to verify that the map ρ := d − δ is a left R-module σ-homomorphism. For the proof of Theorem 1.2 we also need the following two facts. bn ∈ Qmr and c, c1 , . . . , cn ∈ C are elements Lemma 4.3. ∑n Suppose that b1 , . . . ,∑ n satisfying i=1 ci bi = c. Then c ∈ i=1 Cε(bi )ci . ∑n Proof. On the contrary, suppose that c ̸∈ i=1 Cε(bi )ci . By Theorem 2.6 there exists b ∈ R such that cb ̸= 0,

but

Then cb =

n ∑ i=1

ε(bi )ci b = 0 for all 1 ≤ i ≤ n. ci bi b =

n ∑

bi ε(bi )ci b = 0,

i=1

a contradiction.

Proposition 4.4. Let F : R → Qmr , f1 , . . . , fn : R → Qmr be functions satisfying the identity n ∑ F (x)y = fi (x)ybi i=1

for all x, y ∈ R, where b1 , . . . , bn ∈ Qmr . Then there exist central elements c1 , . . . , cn ∈ C such that n ∑ F (x) = ci fi (x) i=1

for all x ∈ R.

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Proof. We assume that F ̸= 0. By Theorem 2.4 there exist elements b′1 , . . . , b′k ∈ ∑n ∑n ⊕k ′ i=1 Cbi such that i=1 Cbi = i=1 Cbi . Then ) ( n n k ∑ ∑ ∑ (4.2) F (x)y = fi (x)ybi = cij fi (x) yb′j , i=1

j=1

i=1

for some central elements cij ∈ C. Hence, if the assertion is true for elements ∑k ⊕k b′1 , . . . , b′k which satisfy i=1 Cb′i = i=1 Cb′i , then (4.2) implies ( ) k n n ∑ ∑ ∑ F (x) = γj cij fi (x) = c′i fi (x) j=1

i=1

i=1

∑k

for all x ∈ R, where c′i := j=1 γj cij ∈ C. Therefore, without loss ⊕ of generality we can assume that elements b1 , . . . , bn ∑n n already satisfy i=1 Cbi = i=i Cbi . Let ε := ε(F (R)) ̸= 0. We have (4.3)

F (x)yε = F (x)y =

n ∑

fi (x)ybi

i=1

for all x, y ∈ R. Note that Lemma 2.7 forces ε ∈ elements γ1 , . . . , γn ∈ C such that n ∑

(4.4)

∑n i=1

Cbi . Therefore, there exist

γi bi = ε.

i=1

Then (4.3) can be rewritten as n ∑ (

) γi F (x) − fi (x) ybi = 0

i=1

for all x, y ∈ R. By Lemma 2.7 and Proposition 2.3 this is equivalent to ( ) (4.5) ε(bi ) γi F (x) − fi (x) = 0 for all 1 ≤ i ≤ n and x ∈ R. Furthermore, by Lemma ∑n 4.3, equality (4.4) implies that there exist elements α1 , . . . , αn ∈ C such that i=1 αi ε(bi )γi = ε. Thus, using (4.5) we get F (x) = F (x)ε =

n ∑

αi ε(bi )γi F (x) =

i=1

n ∑

αi ε(bi )fi (x)

i=1

for all x ∈ R. It remains to define ci := αi ε(bi ) ∈ C for all i = 1, . . . , n.

Proof of Theorem 1.2. (ii) ⇒ (i). This is trivial, since in this case d is implemented by an elementary operator Mp,1 − Ma,a−1 q . (i) ⇒ (ii). We decompose d = δ ∑ + ρ, where δ and ρ are as in Remark 4.2, and n assume that d is implemented by i=1 Mai ,bi . By (1.1) we have δ(x)y = d(xy) − σ(x)d(y) =

n ∑ i=1

(ai x − σ(x)ai )ybi


for all x, y ∈ R. By Proposition 4.4 there exist central elements c1 , . . . , cn ∈ C such that n ∑ δ(x) = ci (ai x − σ(x)ai ) = px − σ(x)p, i=1

∑n

where p := i=1 ci ai . Hence δ is an inner σ-derivation. Furthermore, the map ρ′ : R → Qmr given by ρ′ (x) = d(x) − px = ρ(x) − σ(x)p is a left ∑nR-module σ-homomorphism which is implemented by an elementary operator i=1 Mai ,bi − Mp,1 . By Theorem 3.3 there exist elements q, a ∈ Qmr , where a is invertible, such that ρ′ (x) = σ(x)q and ε(q)σ(x) = ε(q)axa−1 for all x ∈ R. In particular, d(x) =

px + ρ′ (x) = px + σ(x)q

= px + axa−1 q. Finally, note that the proof given above is valid for each pair (p, q) which implements d. This finishes the proof. Theorem 1.2 immediately implies: Corollary 4.5. Let δ : R → Qmr be a σ-derivation, where σ is surjective. Then the following conditions are equivalent: (i) δ is implemented by an elementary operator; (ii) δ is inner, and for each element p ∈ Qmr which implements δ there exists an invertible element a ∈ Qmr such that ε(p)σ(x) = ε(p)axa−1 for all x ∈ R; hence (4.6)

δ = Mp,1 − Ma,a−1 p .

Corollary 4.6. If R is a prime ring, and σ is surjective, then a non-zero σderivation δ : R → Qmr is implemented by an elementary operator if and only if δ is inner σ-derivation, and σ is an X-inner automorphism of R. Remark 4.7. In view of Corollary 4.6, we note that for semiprime rings (which are not prime) σ is not necessarily an X-inner automorphism. For example, let Mn (n ≥ 2) be a ring of all complex n × n matrices, and let R := Mn ⊕ Mn ⊕ Mn . Since each right ideal of Mn is of the form pMn for some projection p ∈ Mn , we have Qmr (Mn ) = Mn , and hence Qmr (R) = R. For 1 ≤ i ≤ 3, let εi be the central idempotent of R with one non-zero entry 1 at i-th coordinate, and let a be a non-central invertible matrix in Mn . We define maps σ, δ : R → R by σ(x) = σ(x1 , x2 , x3 ) := (ax1 a−1 , x3 , x2 ) and

δ(x) := ε1 x − σ(x)ε1 .

Obviously σ is an automorphism of R and δ is a non-zero inner σ-derivation which is implemented by an elementary operator Mε1 ,1 − Ma.ε1 ,a−1 .ε1 . However, σ is not an (X-)inner automorphism of R since σ is not the identity on the center of R = Qmr (R) (for example, σ(ε2 ) = ε3 ).

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At the end of this paper we note that if a σ-derivation δ : R → Qmr , where σ is surjective, is implemented by an elementary operator ψ, then ℓ(ψ) ≤ 2. Indeed, this follows directly from representation (4.6). We now show that ℓ(ψ) = 2 if δ is non-zero: Corollary 4.8. If σ is surjective, then a non-zero σ-derivation δ : R → Qmr cannot be implemented by an elementary operator of length 1. Proof. Assume that δ is implemented by Mu,v for some u, v ∈ Qmr . By Theorem 1.2, there exist elements a, p ∈ Qmr , where a is invertible, such that δ(x) = px − σ(x)p and ε(p)σ(x) = ε(p)axa−1 for all x ∈ R. By the proof of Theorem 1.2 we can assume that p ∈ Cu. Let I be a dense right ideal of R such that pI ⊆ R. Then for all x ∈ R and y ∈ I we have δ(x)δ(y)

= (px − σ(x)p)uyv = (ux − σ(x)u)pyv = δ(xpy) − σ(x)δ(py) = δ(x)py,

which implies δ(x)σ(y)p = 0. Hence, δ(R)aIa−1 p = δ(R)σ(I)p = 0 and so Proposition 2.3 implies ε(p)δ(R) = 0. Therefore δ(R) = ε(p)δ(R) = 0, a contradiction. References [1] P. Ara and M. Mathieu, Local Multipliers of C ∗ -algebras, Springer, London, 2003. [2] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with generalized identities, Marcel Dekker, Inc., 1996. [3] M. Breˇsar, On the distance of the composition of two derivations to the generalized derivation, Glasgow Math. J. 33 (1991), 89–93. [4] T. S. Erickson, W. S. Martindale, 3rd, and J. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), 49-63. [5] I. Gogi´ c, Derivations which are inner as completely bounded maps, Oper. Matrices 4 (2010), 193–211. c, On derivations and elementary operators on C ∗ -algebras, to appear in Proc. Edin. [6] I. Gogi´ Math. Soc. [7] T.-K. Lee, Generalized skew derivations characterized by acting on zero products. Pacific J. Math. 216 (2004), 293–301. [8] G. K. Pedersen, Approximating derivations on ideals of C ∗ -algebras, Invent. Math. 45 (1978), 299–305. [9] Y. Utumi, On quotient rings, Osaka J. Math. 8 (1956), 1–18. Department of Mathematics and Computer Science, FNM, University of Maribor, 2000 Maribor, Slovenia E-mail address: [email protected] ˇka 30, 10000 Zagreb, CroaDepartment of Mathematics, University of Zagreb, Bijenic tia, and Department of Mathematics and Informatics, University of Novi Sad, Trg á 4, 21000 Novi Sad, Serbia Dositeja Obradovic E-mail address: [email protected] ˇka 30, 10000 Zagreb, CroaDepartment of Mathematics, University of Zagreb, Bijenic tia E-mail address: [email protected]