Characterizations of additive $\xi$-Lie derivations on unital algebras
UDC 512.5 Let $\mathscr{R}$ be a commutative ring with unity and $\mathscr{U}$ be a unital algebra over $\mathscr{R}$ (or field $\mathbb{F}$).An $\mathscr{R}$-linear map $L:\mathscr{U}\rightarrow\mathscr{U}$ is called a Lie derivation on $\mathscr{U}$ if $L([u,v])=[L(u),v]+[u,L(v)]$ holds for all $u...
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2021
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507221061795840 |
|---|---|
| author | Ashraf, M. Jabeen, A. Ashraf, M. Jabeen, A. |
| author_facet | Ashraf, M. Jabeen, A. Ashraf, M. Jabeen, A. |
| author_sort | Ashraf, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:48:15Z |
| description | UDC 512.5
Let $\mathscr{R}$ be a commutative ring with unity and $\mathscr{U}$ be a unital algebra over $\mathscr{R}$ (or field $\mathbb{F}$).An $\mathscr{R}$-linear map $L:\mathscr{U}\rightarrow\mathscr{U}$ is called a Lie derivation on $\mathscr{U}$ if $L([u,v])=[L(u),v]+[u,L(v)]$ holds for all $u,$ $v \in\mathscr{U}.$ For scalar $\xi\in\mathbb{F},$ an additive map $L\colon \mathscr{U}\rightarrow\mathscr{U}$ is called an additive $\xi$-Lie derivation on $\mathscr{U}$ if $L([u,v]_{\xi})=[L(u),v]_{\xi}+[u,L(v)]_{\xi},$ where $[u,v]_{\xi}=uv-\xi vu$ holds for all $u, v\in\mathscr{U}.$ In the present paper, under certain assumptions on $\mathscr{U}$it is shown that every Lie derivation (resp., additive $\xi$-Lie derivation) ${L}$ on $\mathscr{U}$ is of standard form, i.e., $L=\delta+\phi,$ where $\delta$ is an additive derivation on $\mathscr{U}$ and $\phi$ is a mapping $\phi\colon \mathscr{U}\rightarrow Z(\mathscr{U})$ vanishing at $[u,v]$ with $uv=0$ in $\mathscr{U}.$ Moreover, we also characterize the additive $\xi$-Lie derivation for $\xi\neq 1$ by its action at zero product in a unital algebra over $\mathbb{F}.$ |
| doi_str_mv | 10.37863/umzh.v73i4.838 |
| first_indexed | 2026-03-24T02:05:52Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i4.838
UDC 512.5
M. Ashraf (Aligarh Muslim Univ., India),
A. Jabeen (Jamia Millia Islamia, India)
CHARACTERIZATIONS OF ADDITIVE \bfitxi -LIE DERIVATIONS
ON UNITAL ALGEBRAS
ХАРАКТЕРИЗАЦIЯ АДИТИВНОГО \bfitxi -ДИФЕРЕНЦIЮВАННЯ ЛI
НА УНIТАЛЬНИХ АЛГЕБРАХ
Let R be a commutative ring with unity and U be a unital algebra over R (or field \BbbF ). An R-linear map L : U \rightarrow U is called
a Lie derivation on U if L([u, v]) = [L(u), v] + [u, L(v)] holds for all u, v \in U. For scalar \xi \in \BbbF , an additive map L :
U \rightarrow U is called an additive \xi -Lie derivation on U if L([u, v]\xi ) = [L(u), v]\xi +[u, L(v)]\xi , where [u, v]\xi = uv - \xi vu holds
for all u, v \in U. In the present paper, under certain assumptions on U it is shown that every Lie derivation (resp., additive
\xi -Lie derivation) L on U is of standard form, i.e., L = \delta +\phi , where \delta is an additive derivation on U and \phi is a mapping \phi :
U \rightarrow Z(U) vanishing at [u, v] with uv = 0 in U. Moreover, we also characterize the additive \xi -Lie derivation for \xi \not = 1
by its action at zero product in a unital algebra over \BbbF .
Нехай R — комутативне кiльце з одиницею, а U — унiтальна алгебра над R (або полем \BbbF ). R-лiнiйне вiдображення
L : U \rightarrow U називається диференцiюванням Лi на U, якщо L([u, v]) = [L(u), v] + [u, L(v)] виконується для всiх u,
v \in U. Для скаляра \xi \in \BbbF адитивне вiдображення L : U \rightarrow U називається адитивним \xi -диференцiюванням Лi на
U, якщо L([u, v]\xi ) = [L(u), v]\xi + [u, L(v)]\xi , де [u, v]\xi = uv - \xi vu виконується для всiх u, v \in U. У цiй роботi при
деяких припущеннях на U доведено, що кожне диференцiювання Лi (вiдповiдно, адитивне \xi -диференцiювання Лi)
L на U має стандартний вигляд, тобто L = \delta +\phi , де \delta — адитивне диференцiювання на U, а \phi — вiдображення \phi :
U \rightarrow Z(U), що зникає на [u, v], якщо uv = 0 у U. Бiльш того, охарактеризовано адитивне \xi -диференцiювання Лi
для \xi \not = 1 через його дiю на нульовий добуток в унiтальнiй алгебрi над \BbbF .
1. Introduction. Throughout, let R be a commutative ring with unity and U be a unital algebra
over R with the center Z(U). For any u, v \in U, [u, v] will denote the commutator uv - vu,
while u \circ v will represent the anticommutator uv + vu. An R-linear map L : U \rightarrow U is called
a derivation (resp., Jordan derivation) on U if L(uv) = L(u)v + uL(v) (resp., L(uv + vu) =
= L(u)v + uL(v) + L(v)u+ vL(u)) holds for all u, v \in U. An R-linear map L : U \rightarrow U is called
a Lie derivation on U if L([u, v]) = [L(u), v] + [u, L(v)] holds for all u, v \in U. Obviously, every
derivation is a Jordan derivation and Lie derivation but not conversely (see [1, 2]).
During the recent past there has been a great deal of work concerning characterization of different
linear mappings viz., Lie derivation, additive \xi -Lie derivation, generalized Lie derivation on various
algebras (see [5, 6, 8 – 14] and references therein). In most of the cases, the object of the studies is
to obtain the conditions under which derivations (Lie derivations) can be completely determined by
the action on some subsets of the algebras. There are several papers on the study of local actions
of Lie derivations of operator algebras. Lu and Jing [8] proved that if X is Banach space of dimension
greater then two and a linear map L : \scrB (X) \rightarrow \scrB (X) such that L([u, v]) = [L(u), v] + [u, L(v)]
for all u, v \in \scrB (X) with uv = 0, than there exists an operator r \in \scrB (X) and a linear map \phi :
\scrB (X) \rightarrow \BbbC I vanishes at all the commutators [u, v] with uv = 0 such that L(u) = ru - ur+\phi (u) for
all u \in \scrB (X). Inspired by this result, Ji and Qi [5] proved that under certain restrictions on triangular
algebra \scrT over commutative ring R, if L : \scrT \rightarrow \scrT is an R-linear map such that L([u, v]) =
= [L(u), v] + [u, L(v)] for all u, v \in \scrT with uv = 0, then there exists a derivation \delta : \scrT \rightarrow \scrT
c\bigcirc M. ASHRAF, A. JABEEN, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4 455
456 M. ASHRAF, A. JABEEN
and an R-linear map \phi : \scrT \rightarrow Z(\scrT ) vanishes at all the commutators [u, v] with uv = 0 such that
L = \delta +\phi . Also, Ji et al. [6] studied the same result on factor von Neumann algebra with dimension
greater than 4. Qi and Hou [12] gave the characterization of Lie derivation on any von Neumann
algebra U without central summands of type I1 and obtained that if L : U \rightarrow U is a linear map such
that L([u, v]) = [L(u), v] + [u, L(v)] for all u, v \in U with uv = 0, then there exists a derivation \delta :
U \rightarrow U and a linear map \phi : U \rightarrow Z(U) vanishes at all the commutators [u, v] with uv = 0 such
that L = \delta + \phi . In particular, L : U \rightarrow U is a linear map such that L([u, v]) = [L(u), v] + [u, L(v)]
for all u, v \in U with uv = 0, if and only if there exists an operator r \in U and a linear map \phi :
U \rightarrow Z(U) vanishes at all the commutators [u, v] with uv = 0 such that L(u) = ur - ru + \phi (u)
for all u \in U. Furthermore, Qi [13] characterized Lie derivation on \scrJ -subspace lattice algebras
and proved the same result due to Lu and Jing [8] on \scrJ -subspace lattice algebra \mathrm{A}\mathrm{l}\mathrm{g}\scrL , where \scrL
is \scrJ -subspace lattice on a Banach space X over the real or complex field with dimension greater
than 2.
Let U be a unital algebra over real or complex field \BbbF . If any pair u, v \in U commute, then
their Lie product is zero. For scalar \xi \in \BbbF and u, v \in U, u commutes with v up to a factor \xi if
uv = \xi vu. In the theme of quantum groups and operator algebras [3, 7], the notion of commutativity
up to a factor for pairs of operators has been studied. Qi and Hou [9] introduced the concept of \xi -Lie
derivation. For any u, v \in U, [u, v]\xi = uv - \xi vu will denote the \xi -Lie product. A linear map L :
U \rightarrow U is said to be a \xi -Lie derivation if L([u, v]\xi ) = [L(u), v]\xi + [u, L(v)]\xi . If L is additive, then
\xi -Lie derivation is called an additive \xi -Lie derivation. It can be easily seen that if \xi = 0, 1, - 1, then
\xi -Lie derivation is called derivation, Lie derivation and Jordan derivation, respectively. Note that an
additive map L : U \rightarrow U is called a generalized derivation if L(uv) = L(u)v+ uL(v) - uL(I)v for
all u, v \in U. During the recent years many authors characterized \xi -Lie derivation on several rings
and operator algebras (see [10, 12, 14]). Characterization of \xi -Lie derivation on prime algebras, von
Neumann algebra and triangular algebra can be found in [10, 14, 15].
Motivated by the above observations, in Section 3, we characterize a Lie derivation on unital
algebras over a commutative ring R at zero product and prove that if L : U \rightarrow U is an R-linear
map such that L([u, v]) = [L(u), v] + [u, L(v)] for all u, v \in U with uv = 0, then under certain
appropriate restrictions on U there exists a derivation \delta : U \rightarrow U and an R-linear map \phi : U \rightarrow Z(U)
vanishes at all the commutators [u, v] with uv = 0 such that L = \delta + \phi . In Section 4, we study
the characterization of \xi -Lie derivation at zero product on unital algebras over a field \BbbF with certain
limitations and find that an additive map satisfies L([u, v]\xi ) = [L(u), v]\xi +[u, L(v)]\xi for all u, v \in U
with uv = 0 if and only if L(I) \in Z(U) and (i) for \xi \not = 0, - 1, L(\xi uv) = \xi L(u)v + \xi uL(v) and
there exists an additive derivation \delta satisfying \delta (\xi I) = \xi L(I) such that L(u) = \delta (u)+L(I)u for all
u \in U; (ii) for \xi = - 1, L is an additive derivation; (iii) for \xi = 0, there exists an additive derivation
\delta such that L(u) = \delta (u) + L(I)u for all u \in U. In the last section, we discuss some applications of
these results on few important examples of unital algebras.
2. Preliminaries. Let U be an unital algebra over a commutative ring R with an idempotent
p \not = 0 and let q = 1 - p. Then according to the well known Peirce decomposition formula, U can
be represented as U = pUp + pUq + qUp + qUq, where pUp and qUq are subalgebras with unital
elements p and q, respectively, pUq is an (pUp, qUq)-bimodule and qUp is an (qUq, pUp)-bimodule.
We will assume that U satisfies
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CHARACTERIZATIONS OF ADDITIVE \xi -LIE DERIVATIONS . . . 457
pup.pUq = \{ 0\} = qUp.pup implies pup = 0,
pUq.quq = \{ 0\} = quq.qUp implies quq = 0
(2.1)
for all u \in U. Some specific examples of unital algebras with nontrivial idempotents having the
property (2.1) are triangular algebras, matrix algebras, algebras of all bounded linear operators of
Banach space and the unital prime algebras with nontrivial idempotents.
Throughout, this paper we shall use the following notions: Let U = pUp + pUq + qUp + qUq
be unital algebra with nontrivial idempotents p and q = 1 - p satisfying (2.1). Let U11 = pUp,
U12 = pUq, U21 = qUp and U22 = qUp. Then U = U11 + U12 + U21 + U22. The center of U is
Z(U) = \{ u11 + u22 \in U11 + U22 | u11u12 = u12u22, u21u11 = u22u21 \forall u12 \in U12, u21 \in U21\} .
Define two natural projections \pi U11 : U \rightarrow U11 and \pi U22 : U \rightarrow U22 by
\pi U11(u11 + u12 + u21 + u22) = u11 and \pi U22(u11 + u12 + u21 + u22) = u22.
Moreover, \pi U11(Z(U)) \subseteq Z(U11) and \pi U22(Z(U)) \subseteq Z(U22) and there exists a unique algebra
isomorphism \tau : \pi U11(Z(U)) \rightarrow \pi U22(Z(U)) such that u11u12 = u12\tau (u11) and u21u11 = \tau (u11)u21
for all u11 \in \pi U11(Z(U)), u12 \in U12, u21 \in U21.
3. Characterization of Lie derivations. In this section, we characterize Lie derivation by action
at zero product on a unital algebra with a nontrivial idempotent. Actually, we prove the following
result.
Theorem 3.1. Let U be a 2-torsion free unital algebra over a commutative ring R with a
nontrivial idempotent p satisfying (2.1), \pi U11(Z(U)) = Z(U11) and \pi U22(Z(U)) = Z(U22). If L :
U \rightarrow U is an R-linear mapping satisfying L([u, v]) = [L(u), v] + [u, L(v)] for all u, v \in U with
uv = 0, there exists a derivation \delta : U \rightarrow U and an R-linear map \phi : U \rightarrow Z(U) vanishing at all
the commutators [u, v] with uv = 0 such that L = \delta + \phi .
Throughout this section we assume that U is 2-torsion free unital algebra over a commutative
ring R satisfying the hypotheses of Theorem 3.1. In order to prove the above result, we start with
the following sequence of lemmas.
Lemma 3.1. For any p, q:
(i) pL(p)p+ qL(p)q \in Z(U),
(ii) pL(q)p+ qL(q)q \in Z(U),
(iii) pL(I)p+ qL(I)q \in Z(U).
Proof. (i) Since u12p = 0 for any u12 \in U12, it follows that
- L(u12) = [L(u12), p] + [u12, L(p)] = L(u12)p - pL(u12) + u12L(p) - L(p)u12. (3.1)
Multiplying the last equality by p and q from left and right, respectively, we find u12qL(p)q =
= pL(p)pu12. Also, pu21 = 0 implies that qL(p)qu21 = u21pL(p)p. From the last two expressions
we arrive at pL(p)p+ qL(p)q \in Z(U).
(ii) Since u21q = 0 = qu12, using similar steps as used in (i), we obtain that pL(q)p+ qL(q)q \in
\in Z(U).
(iii) Since p(I - p) = 0, we have
0 = [L(p), I - p] + [p, L(I - p)] = pL(I) - L(I)p.
This implies that pL(I)q = 0 = qL(I)p. Now, from (i) and (ii), we get L(I) = pL(I)p+ qL(I)q \in
\in Z(U).
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458 M. ASHRAF, A. JABEEN
Remark 3.1. Define a map L\prime : U \rightarrow U by L\prime (u) = L(u) + [u0, u] for all u \in U, where
u0 = pL\prime (p)q - qL\prime (p)p.
Accordingly, consider only those Lie derivation L : U \rightarrow U which satisfies pL(p)q = 0 =
= qL(p)p. Now, it can be easily seen from the following lemma.
Lemma 3.2. For any u, v \in U :
(i) L([u, v]) = [L(u), v] + [u, L(v)],
(ii) L(I), L(p), L(q) \in Z(U).
Lemma 3.3. For any uij \in Uij , L(uij) \in Uij , where i \not = j \in \{ 1, 2\} .
Proof. Consider the case for i = 1 and j = 2. As u12p = 0, using (3.1) we find that
pL(u12)p = qL(u12)p = qL(u12)q = 0. Hence, L(u12) \in U12. Similarly, we can calculate for
i = 2 and j = 1.
Lemma 3.4. L(Uii) \subseteq U11 \oplus U22, and there exists a map \phi i : Uii \rightarrow Z(U) such that L(uii) -
- \phi i(uii) \in Uii for all uii \in Uii, i = 1, 2.
Proof. Since u11q = 0 for any u11 \in U11 , it follows that
0 = [L(u11), q] + [u11, L(q)] = L(u11)q - qL(u11).
This yields that L(u11) \in U11 + U22. Now for any vii \in Uii, i = 1, 2, we can write L(u11) =
= v11 + v22. Since u11u22 = 0, we obtain
0 = [L(u11), u22] + [u11, L(u22)] = v22u22 - u22v22 + [u11, L(u22)].
Now multiplying both sides by q in the above expression, it follows that v22 \in Z(U22). Thus, for
any z \in Z(U),
L(u11) = v11 + zq = v11 + z - zp = (v11 - zp) + z \in U11 + Z(U).
Hence we conclude that there exists a map \phi 1 : U11 \rightarrow Z(U) such that L(u11) - \phi 1(u11) \in U11 for
all u11 \in U11. Since L is R-linear map, we can easily see that \phi 1 is R-linear map. Similarly, we
can show the result for i = 2.
Remark 3.2. Now we define two R-linear maps \phi : U \rightarrow Z(U) and \delta : U \rightarrow U by \phi (u) =
= \phi 1(pup) + \phi 2(quq) and \delta (u) = L(u) - \phi (u) for all u \in U. It is clear that \delta (uij) \in Uij for 1 \leq i,
j \leq 2 and \delta (uij) = L(uij) for 1 \leq i \not = j \leq 2.
Lemma 3.5. For any uij \in Uij , 1 \leq i, j \leq 2 :
(i) \delta (uiiuij) = \delta (uii)uij + uii\delta (uij), where i \not = j \in \{ 1, 2\} ,
(ii) \delta (ujiuii) = \delta (uji)uii + uji\delta (uii), where i \not = j \in \{ 1, 2\} .
Proof. (i) Since u12u11 = 0, for any u11 \in U11 and u12 \in U12, we have
- \delta (u11u12) = - L(u11u12) =
= [L(u12), u11] + [u12, L(u11)] =
= L(u12)u11 - u11L(u12) + u12L(u11) - L(u11)u12 =
= - \delta (u11)u12 - u11\delta (u12).
This gives that \delta (u11u12) = \delta (u11)u12 + u11\delta (u12) for all u11 \in U11 and u12 \in U12.
In a similar manner, we can obtain rest of the cases.
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CHARACTERIZATIONS OF ADDITIVE \xi -LIE DERIVATIONS . . . 459
Lemma 3.6. For any uii, vii \in Uii, i = 1, 2 :
(i) \delta (uiivii) = \delta (uii)vii + uii\delta (vii),
(ii) \delta (p) = 0 = \delta (q).
Proof. Consider the case for i = 1. For any t12 \in U12, using Lemma 3.5 we find \delta (u11v11t12) =
= \delta (u11v11)t12 + u11v11\delta (t12).
On the other hand, \delta (u11v11t12 = \delta (u11)v11t12 + u11v11\delta (t12) + u11\delta (v11)t12. From the above
two expressions, we have \{ \delta (u11v11) - \delta (u11)v11 - u11\delta (v11)\} t12 = 0. Similarly, by using Lemma
3.6, we obtain t21\{ \delta (u11v11) - \delta (u11)v11 - u11\delta (v11)\} = 0. Now by (2.1) we arrive at \delta (u11v11) =
= \delta (u11)v11 + u11\delta (v11) for all u11, v11 \in U11. Similarly for i = 2.
(ii) From (i) and using Remark 3.2, we find \delta (p) = 0 = \delta (q).
Lemma 3.7. For any uij , vij \in Uij , \delta (uijvji) = \delta (uij)vji + uij\delta (vji), where i \not = j \in \{ 1, 2\} .
Proof. First consider i = 1 and j = 2. Since (u12v21 - u12 - v21 + q)(p + v21) = 0 for any
u12 \in U12 and v21 \in U21 , we have
- \delta (u12v21) + \delta (u12) - \delta (v21u12v21) + \delta (v21u12) = - \delta (u12v21 - u12 + v21u12v21 - v21u12) =
= \delta ([u12v21 - u12 - v21 + q, p+ v21]) =
= [L(u12v21) - L(u12) - L(v21) + L(q), p+ v21]+
+[u12v21 - u12 - v21 + q, L(p) + L(v21)] =
= \delta (u12) - \delta (u12)v21 - v21\delta (u12v21) + v21\delta (u12) -
- u12\delta (v21) - \delta (v21)u12v21 + \delta (v21)u12.
By using Lemma 3.5, it follows that
- \delta (u12)v21 + v21\delta (u12) - u12\delta (v21) + \delta (v21)u12 + \delta (u12v21) - \delta (v21u12) = 0. (3.2)
Multiplying the above expression by u12 from right and u21 from left, respectively, we obtain
\{ - \delta (u12)v21 - u12\delta (v21) + \delta (u12v21)\} u12 = 0 and u21\{ - \delta (u12)v21 - u12\delta (v21) + \delta (u12v21)\} = 0.
By assumption, we have \delta (u12v21) = \delta (u12)v21 + u12\delta (v21) for all u12 \in U12 and v12 \in U21.
In the similar manner, multiplying (3.2) by u21 from the right and u12 from the left, respectively,
we obtain for i = 1 and j = 2.
Proof of Theorem 3.1. In view of Remark 3.2 it remains to show that \delta (uv) = \delta (u)v + u\delta (v)
for all u, v \in U. Suppose that u = u11 + u12 + u21 + u22 and v = v11 + v12 + v21 + v22 for all u11,
v11 \in U11; u12, v12 \in U12; u21, v21 \in U21 and u22, v22 \in U22. Now, using Lemmas 3.6 and 3.7 it
follows that
\delta (uv) = \delta ((u11 + u12 + u21 + u22)(v11 + v12 + v21 + v22)) =
= \delta (u11v11) + \delta (u11v12) + \delta (u12v21) + \delta (u12v22)+
+\delta (u21v11) + \delta (u21v12) + \delta (u22v21) + \delta (u22v22) =
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460 M. ASHRAF, A. JABEEN
= \delta (u11)v11 + \delta (u11)v12 + \delta (u12)v21 + \delta (u12)v22 + \delta (u21)v11+
+\delta (u21)v12 + \delta (u22)v21 + \delta (u22)v22 + u11\delta (v11) + u11\delta (v12)+
+u12\delta (v21) + u12\delta (v22) + u21\delta (v11) + u21\delta (v12) + u22\delta (v21) + u22\delta (v22) =
= \delta (u11 + u12 + u21 + u22)(v11 + v12 + v21 + v22)+
+(u11 + u12 + u21 + u22)\delta (v11 + v12 + v21 + v22) =
= \delta (u)v + u\delta (v).
That is, \delta is a derivation on U. Lastly, we have to show that \phi [u, v] = 0 with uv = 0:
\phi ([u, v]) = [L(u), v] + [u, L(v)] - \delta ([u, v]) = [\delta (u), v] + [u, \delta (v)] - \delta ([u, v]) = 0.
Therefore, Lie derivation L has standard form, i.e., L can be written as a sum of derivation and a
linear map vanishing at commutator by the action at zero product.
Theorem 3.1 is proved.
4. Characterization of \xi -Lie derivations. In this section, we characterize \xi -Lie derivations for
\xi \not = 1 by its action at zero product on a unital algebra containing nontrivial idempotents.
Theorem 4.1. Suppose that U is a 2-torsion free unital algebra over a field \BbbF with a nontrivial
idempotent p satisfying (2.1) and \pi U11(Z(U)) = Z(U11), \pi U22(Z(U)) = Z(U22). Let L : U \rightarrow U
be an additive mapping satisfying L([u, v]\xi ) = [L(u), v]\xi + [u, L(v)]\xi for all u, v \in U with uv = 0.
Then we have the following cases:
(i) If \xi \not = 0, - 1, then L(\xi uv) = \xi L(u)v + \xi uL(v) and there exists an additive derivation \delta
satisfying \delta (\xi I) = \xi L(I) such that L(u) = \delta (u) + L(I)u if only if L(I) \in Z(U) for all u \in U.
(ii) If \xi = - 1, then L is an additive derivation.
(iii) If \xi = 0, then there exists an additive derivation \delta such that L(u) = \delta (u) + L(I)u if and
only if L(I) \in Z(U) for all u \in U.
Assume that U is 2-torsion free unital algebra over a field \BbbF satisfying the hypotheses of Theo-
rem 4.1. The direct part is obvious. To prove only if part, we need the following lemmas.
Lemma 4.1. pL(I)q = 0 = qL(I)p and qL(p)q = 0 = pL(q)p.
Proof. Since pq = 0, we find
0 = L([p, q]\xi ) = L(p)q - \xi qL(p) + pL(q) - \xi L(q)p. (4.1)
Now multiplying by q on both side of (4.1), we get qL(p)q = 0. Again, multiplying by p and q
from left and right, respectively, in (4.1), we obtain pL(p)q+ pL(q)q = pL(I)q = 0. As qp = 0, on
the similar steps we can find that qL(I)p = 0 and pL(q)p = 0.
Remark 4.1. Define a map L\prime : U \rightarrow U by L\prime (u) = L(u) + [u0, u] for all u \in U, where u0 =
= pL\prime (p)q - qL\prime (p)p.
Accordingly, consider only those \xi -Lie derivation L : U \rightarrow U which satisfies pL(p)q = 0 =
qL(p)p. Now, it can be easily seen from the following lemma.
Lemma 4.2. L(p) = pL(p)p and L(q) = qL(q)q.
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CHARACTERIZATIONS OF ADDITIVE \xi -LIE DERIVATIONS . . . 461
Lemma 4.3. For any uij \in Uij , i \not = j and i, j \in \{ 1, 2\} :
(i) L(I) \in Z(U),
(ii) L(uij) \in Uij .
Proof. (i) Since u12p = 0 for all u12 \in U12, we have
L( - \xi u12) = L([u12, p]\xi ) = L(u12)p - \xi pL(u12) - \xi L(p)u12. (4.2)
Since qu12 = 0, using Lemma 4.2, we obtain
L( - \xi u12) = - \xi u12L(q) + qL(u12) - \xi L(u12)q. (4.3)
Combining (4.2) and (4.3), we arrive at
L(u12)p - \xi pL(u12) - \xi L(p)u12 = - \xi u12L(q) + qL(u12) - \xi L(u12)q. (4.4)
Now, if \xi \not = 0, then by multiplying p and q on left and right, respectively, in (4.4) and using
Lemma 4.2, we have L(p)u12 = pL(p)pu12 = u12qL(q)q = u12L(q).
Now, in case of \xi = 0, using (p+ u12)(u12 - q) = 0 = (u12 - q)(p+ u12), we find
0 = L(p+ u12)(u12 - q) + (p+ u12)L(u12 - q) =
= L(p)u12 + L(u12)u12 - L(u12)q + pL(u12) + u12L(u12) - u12L(q)
and
0 = L(u12 - q)(p+ u12) + (u12 - q)L(p+ u12) =
= L(u12)p+ L(u12)u12 + u12L(u12) - qL(u12).
From the above two expressions, we arrive at L(p)u12 = u12L(q). We have
L(I)u12 = L(p+ q)u12 = L(p)u12 = u12L(q) + u12L(p) = u12L(I).
Also, we can show that L(q)u21 = u21L(p) and hence L(I)u21 = u21L(I). This implies that
L(I) \in Z(U).
(ii) Now multiplying by p and q, respectively, on both the sides of (4.4), we find pL(u12)p =
= 0 = qL(u12)q.
If \xi = 0, then from (4.2), we get qL(u12)p = 0 which leads to L(u12) \in U12.
If \xi = - 1, then on using (p+ u12)(u12 - q) = 0 and by (4.4), we obtain
0 = L(p+ u12)(u12 - q) + (p+ u12)L(u12 - q) + L(u12 - q)(p+ u12) + (u12 - q)L(p+ u12) =
= L(p)u12 + 2L(u12)u12 - L(u12)q + 2u12L(u12) =
= - qL(u12) + L(u12)p+ pL(u12) - u12L(q) =
= 2L(u12)u12 + 2u12L(u12).
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462 M. ASHRAF, A. JABEEN
This yields that qL(u12)pu12 = 0 = u12qL(u12)p and hence qL(u12)p = 0.
If \xi \not = 0, - 1, then for any u12, v12 \in U12 using the fact (v12 - q)(p+ u12) = 0, we obtain
L(\xi u12 - \xi v12) = L([v12 - q, p+ u12]\xi ) =
= L(v12)p+ L(v12)u12 - \xi pL(v12) - \xi u12L(v12) + \xi u12L(q)+
+v12L(u12) - qL(u12) - \xi L(p)v12 - \xi L(u12)v12 + \xi L(u12)q. (4.5)
Now multiplying (4.5) by p on both sides and using the fact pL(u12)p = 0, we get \xi u12qL(v12)p =
= v12qL(u12)p. Multiplying (4.2) by q on the left-hand side and p on the right-hand side, we have
qL( - \xi u12)p = qL(u12)p. Combining the above two equations, we obtain
v12qL(u12)p = v12qL( - \xi u12)p = - \xi 2u12qL(v12)p = - \xi v12qL(u12)p.
This implies that qL(u12)p = 0 and hence L(u12) \in U12 for all u12 \in U12. In the similar manner,
we can prove that L(u21) \in U21 for all u21 \in U21.
Lemma 4.4. For any uii \in Uii, L(uii) \in Uii, where i = 1, 2.
Proof. Since u11q = 0 for any u11 \in U11, we have
0 = [L(u11), q]\xi + [u11, L(q)]\xi =
= L(u11)q - \xi qL(u11) + u11L(q) - \xi L(q)u11 =
= L(u11)q - \xi qL(u11). (4.6)
Now using the fact \xi \not = 1 and multiplying (4.6) by q on both sides, by p on left- and right-hand side,
respectively, we get qL(u11)q = pL(u11)q = qL(u11)p = 0. This implies that L(u11) = pL(u11)p
for all u11 \in U11.
If \xi = 0, then from (4.6) we obtain pL(u11)q = 0 = qL(u11)q. Note that qu11 = 0 which gives
that
0 = [q, L(u11)]\xi + [L(q), u11]\xi =
= qL(u11) - \xi L(u11)q + L(q)u11 - \xi u11L(q) =
= qL(u11) - \xi L(u11)q.
This implies that qL(u11)p = 0 and hence L(u11) \in U11 for all u11 \in U11. Similarly, we can show
for i = 2.
Proof of Theorem 4.1. The proof is divide in following two steps:
Step 1. The following statements are true:
(i) If \xi \not = 0, - 1, then L(\xi xy) = \xi L(x)y + \xi xL(y) for all x, y \in \scrA .
(ii) If \xi = - 1, then L is an additive derivation.
(iii) If \xi = 0, then there exists an additive derivation \delta such that L(x) = \delta (x) + L(I)x for all
x \in \scrA .
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CHARACTERIZATIONS OF ADDITIVE \xi -LIE DERIVATIONS . . . 463
(i) Since yijxii = 0 for all xii \in \scrA ii, yij \in \scrA ij and 1 \leq i \not = j \leq 2, we obtain
- L(\xi xiiyij) = [L(yij), xii]\xi + [yij , L(xii)]\xi = - \xi xiiL(yij) - \xi L(xii)yij .
This implies that
L(xiiyij) = \xi L(xii)yij + \xi xiiL(yij). (4.7)
Similarly, we can find
L(xijyjj) = \xi L(xij)yjj + \xi xijL(yjj). (4.8)
Now, for any xii, yii \in \scrA ii, we have L(xiiyiiyij) = \xi L(xiiyii)yij + \xi xiiyiiL(yij).
On the other hand, L(xiiyiiyij) = \xi L(xii)yiiyij + \xi 2xiiL(\xi
- 1yii)yij + \xi xiiyiiL(yij). Combining
the above two expressions, we get
0 = (L(xiiyii) - L(xii)yii - \xi xiiL(\xi
- 1yii))yij =
= (L(\xi xiiyii) - \xi L(xii)yii - \xi xiiL(yii))yij .
Similarly, using (4.8) we find yji(L(\xi xiiyii) - \xi L(xii)yii - \xi xiiL(yii)) = 0. Now by assumption,
the last two expressions leads to
L(\xi xiiyii) = \xi L(xii)yii + \xi xiiL(yii). (4.9)
Again for all xij \in \scrA ij , yji \in \scrA ji and i \not = j, (xijyji - xij - yji + q)(p + yji) = 0. Now applying
similar steps as used in proof of Theorem 3.1, we arrive at
L(\xi xijyji) = \xi L(xij)yji + \xi xijL(yji). (4.10)
Applying (4.7) – (4.10) and using the similar calculation as used in proof of Theorem 3.1, we obtain
L(\xi xy) = \xi L(x)y + \xi xL(y) for all x, y \in \scrA .
(ii) If we take \xi = - 1 in (i), then we find that L is an additive derivation.
(iii) Note that if \xi = 0 and xy = 0, then by definition we have L(x)y + xL(y) = 0 for all
x, y \in \scrA . Also, here we will use the fact L(p)x12 = x12L(q) and L(q)x21 = x21L(p) for all
x12 \in \scrA 12 and x21 \in \scrA 21. Since (x11 + x11y12)(q - y12) = 0 for all x11 \in \scrA 11 and y12 \in \scrA 12, we
have
0 = L(x11 + x11y12)(q - y12) + (x11 + x11y12)L(q - y12) =
= - L(x11)y12 + L(x11y12) - x11L(y12) + x11y12L(q).
This implies that
L(x11y12) = L(x11)y12 + x11L(y12) - x11L(p)y12. (4.11)
Also, note that (x22 + x22y21)(p - y21) = (p - x12)(y22 + x12y22) = (q - x21)(y22 + x12y22) = 0.
By using these relations, we obtain
L(x22y21) = L(x22)y21 + x22L(y21) - x22L(q)y21, (4.12)
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464 M. ASHRAF, A. JABEEN
L(x12y22) = L(x12)y22 + x12L(y22) - x12L(p)y22, (4.13)
L(x21y11) = L(x21)y11 + x21L(y11) - x21L(q)y11. (4.14)
Now, using (4.11), (4.14) and applying similar steps after (4.8), we find
L(xiiyii) = L(xii)yii + xiiL(yii) - xiiL(p)yii. (4.15)
Also, since (x12 + x12y21)(p - y21) = 0 = (x21 + x21y12)(q - y12) and using Lemma 4.3, we have
L(x12y21) = L(x12)y21 + x12L(y21) - x12L(p)y21, (4.16)
L(x21y12) = L(x21)y12 + x21L(y12) - x21L(q)y12. (4.17)
Applying (4.11) – (4.17) and using the similar calculation as used in the proof of Theorem 3.1, we
obtain that L(xy) = L(x)y + xL(y) - xL(I)y for all x, y \in \scrA .
Step 2. If \xi \not = 0, - 1, then there exists an additive derivation \delta satisfying \delta (\xi I) = \xi L(I) such
that L(x) = \delta (x) + L(I)x for all x \in \scrA .
By Step 1, we have L(\xi xy) = \xi L(x)y + \xi xL(y). As xy = 0 we find \xi L(x)y + \xi xL(y) = 0
and hence L(x)y + xL(y) = 0. Again, by Lemma 4.4 (Step 2), it is similar to the case \xi = 0 and
hence there exists an additive derivation \delta satisfying L(x) = \delta (x) + L(I)x for all x \in \scrA . Also,
L(\xi I) = \xi L(I)I + \xi IL(I) = 2\xi L(I). Therefore, L(\xi I) \in Z(\scrA ). Now, since x12p = 0,
\delta ( - \xi x12) - \xi L(I)x12 = L( - \xi x12) = [L(x12), p]\xi + [x12, L(p)]\xi =
= [\delta (x12) + L(I)x12, p]\xi + [x12, \delta (p) + L(I)p]\xi = - \xi \delta (x12) - 2\xi L(I)x12.
This gives that \delta (\xi x12) = \xi \delta (x12)+ \xi L(I)x12. On the other hand, \delta (\xi x12) = \delta (\xi I)x12+ \xi \delta (x12). It
follows that
\bigl(
\delta (\xi I) - \xi L(I)
\bigr)
x12 = 0. In the similar way, we can obtain that x21
\bigl(
\delta (\xi I) - \xi L(I)
\bigr)
= 0.
Hence by (2.1), we have \delta (\xi I) = \xi L(I).
Theorem 4.1 is proved.
5. Applications. As a direct consequence of our Theorem 3.1, we have the following results.
Corollary 5.1 ([8], Theorem 2.1). Let X be a Banach space of dimension greater than 2, and L :
\scrB (X) \rightarrow \scrB (X) be a linear map satisfying L([u, v]) = [L(u), v] + [u, L(v)] for any u, v \in \scrB (X)
with uv = 0. Then there exists an operator r \in \scrB (X) and a linear map \phi : \scrB (X) \rightarrow \BbbC I vanishing
at commutators [u, v] when uv = 0 such that L(u) = ru - ur + \phi (u) for all u \in \scrB (X).
Corollary 5.2 ([5], Theorem 2.1). Let \scrA and \scrB be two algebras over a commutative ring R with
unity I1 and I2, respectively. Let \scrM be a faithful (\scrA ,\scrB )-bimodule and \scrT = \mathrm{T}\mathrm{r}\mathrm{i}(\scrA ,\scrM ,\scrB ) be a
triangular algebra consisting of \scrA , \scrM and \scrB . If \pi \scrA (Z(\scrT )) = Z(\scrA ) and \pi \scrB (Z(\scrT )) = Z(\scrB ) and
L : \scrT \rightarrow \scrT is an R-linear map such that L([u, v]) = [L(u), v] + [u, L(v)] for any u, v \in \scrT with
uv = 0, then there exists a derivation \delta of \scrT and an R-linear map \phi : \scrT \rightarrow Z(\scrT ) vanishing at
commutators [u, v] with uv = 0 such that L(u) = \delta (u) + \phi (u) for all u \in \scrT .
Corollary 5.3 ([5], Corollary 2.1). Let \scrN be an arbitrary nest on a Hilbert space H of dimen-
sion greater than 2 and \mathrm{A}\mathrm{l}\mathrm{g}\scrN be the associated nest algebra. Let L : \mathrm{A}\mathrm{l}\mathrm{g}\scrN \rightarrow \mathrm{A}\mathrm{l}\mathrm{g}\scrN be a linear
map satisfying L([u, v]) = [L(u), v]+ [u, L(v)] for any u, v \in \mathrm{A}\mathrm{l}\mathrm{g}\scrN with uv = 0. Then there exists
an operator r \in \mathrm{A}\mathrm{l}\mathrm{g}\scrN and a linear map \phi : \mathrm{A}\mathrm{l}\mathrm{g}\scrN \rightarrow \BbbF I vanishing at commutators [u, v] when
uv = 0 such that L(u) = ru - ur + \phi (u) for all u \in \mathrm{A}\mathrm{l}\mathrm{g}\scrN .
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CHARACTERIZATIONS OF ADDITIVE \xi -LIE DERIVATIONS . . . 465
For the finite dimensional case, it is clear that every nest algebra on a finite dimensional space is
isomorphic to an upper triangular block matrix algebra [4].
Corollary 5.4. Let \BbbF be the real or complex field and n > 2 be a positive integer. Let \scrB n(R) be
a proper block upper triangular matrix algebra over \BbbF and L : \scrB n(R) \rightarrow \scrB n(R) be a linear map
satisfying L([u, v]) = [L(u), v] + [u, L(v)] for any u, v \in \scrB n(R) with uv = 0. Then there exists
an operator r \in \scrB n(R) and a linear map \phi : \scrB n(R) \rightarrow \BbbF I vanishing at commutators [u, v] when
uv = 0 such that L(u) = ru - ur + \phi (u) for all u \in \scrB n(R).
Corollary 5.5. Let U be a factor von Neumann algebra with deg(U) > 1 and L : U \rightarrow U be
a linear map. Then L satisfies L([u, v]) = [L(u), v] + [u, L(v)] for any u, v \in U with uv = 0
if and only if it has the form L(u) = ru - ur + \tau (u) for all u \in U, where r \in U and \tau :
U \rightarrow \BbbF I is a linear functional vanishing on each commutator [u, v] whenever uv = 0.
Proof. Since factor von Neumann algebra U satisfies (2.1) and all linear derivations of von
Neumann algebras are inner, L is the sum of inner derivation and a linear functional vanishing on
each commutator [u, v] whenever uv = 0.
Since every triangular algebra is the example of algebra that satisfies (2.1), the following result
is an immediate consequence of the Theorem 4.1.
Corollary 5.6 ([11], Theorem 4.1). Let \scrA and \scrB be unital algebras over a field \BbbF , and \scrM
be an (\scrA ,\scrB ) bimodule, which is faithful as a left \scrA -module and also as a right \scrB -module. Let
\scrT = \mathrm{T}\mathrm{r}\mathrm{i}(\scrA ,\scrM ,\scrB ) be the triangular algebra consisting of \scrA ,\scrB ,\scrM and \xi \in \BbbF with \xi \not = 0, 1.
Assume that L : \scrT \rightarrow \scrT is an additive map, \pi \scrA (Z(\scrT )) = Z(\scrA ) and \pi \scrB (Z(\scrT )) = Z(\scrB ). Then
L satisfies L([u, v]\xi ) = [L(u), v]\xi + [u, L(v)]\xi for all u, v \in \scrT with uv = 0 if and only if
L(I) \in Z(\scrT ) and there exists an additive derivation \delta : \scrT \rightarrow \scrT with \delta (\xi I) = \xi L(I) such that
L(u) = \delta (u) + L(I)u for all u \in \scrT .
Corollary 5.7 ([11], Theorem 4.2). Let \scrN be a nest on an infinite dimensional Banach space
X over the real or complex field \BbbF , and let \mathrm{A}\mathrm{l}\mathrm{g}\scrN be the associated nest algebra. Assume that
\xi \in \BbbF with \xi \not = 0, 1 and L : \mathrm{A}\mathrm{l}\mathrm{g}\scrN \rightarrow \mathrm{A}\mathrm{l}\mathrm{g}\scrN is an additive map and there exists a nontrivial
element in \scrN which is complemented in X. Then L satisfies L([u, v]\xi ) = [L(u), v]\xi + [u, L(v)]\xi
for any u, v \in \mathrm{A}\mathrm{l}\mathrm{g}\scrN with uv = 0 if and only if there exists an operator r \in \mathrm{A}\mathrm{l}\mathrm{g}\scrN such that
L(u) = ur - ru for all u \in \mathrm{A}\mathrm{l}\mathrm{g}\scrN .
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Received 30.06.17,
after revision — 10.05.19
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| spelling | umjimathkievua-article-8382025-03-31T08:48:15Z Characterizations of additive $\xi$-Lie derivations on unital algebras Characterizations of additive $\xi$-Lie derivations on unital algebras Ashraf, M. Jabeen, A. Ashraf, M. Jabeen, A. unital algebra Lie derivation ξ-Lie derivation unital algebra Lie derivation ξ-Lie derivation UDC 512.5 Let $\mathscr{R}$ be a commutative ring with unity and $\mathscr{U}$ be a unital algebra over $\mathscr{R}$ (or field $\mathbb{F}$).An $\mathscr{R}$-linear map $L:\mathscr{U}\rightarrow\mathscr{U}$ is called a Lie derivation on $\mathscr{U}$ if $L([u,v])=[L(u),v]+[u,L(v)]$ holds for all $u,$ $v \in\mathscr{U}.$ For scalar $\xi\in\mathbb{F},$ an additive map $L\colon \mathscr{U}\rightarrow\mathscr{U}$ is called an additive $\xi$-Lie derivation on $\mathscr{U}$ if $L([u,v]_{\xi})=[L(u),v]_{\xi}+[u,L(v)]_{\xi},$ where $[u,v]_{\xi}=uv-\xi vu$ holds for all $u, v\in\mathscr{U}.$ In the present paper, under certain assumptions on $\mathscr{U}$it is shown that every Lie derivation (resp., additive $\xi$-Lie derivation) ${L}$ on $\mathscr{U}$ is of standard form, i.e., $L=\delta+\phi,$ where $\delta$ is an additive derivation on $\mathscr{U}$ and $\phi$ is a mapping $\phi\colon \mathscr{U}\rightarrow Z(\mathscr{U})$ vanishing at $[u,v]$ with $uv=0$ in $\mathscr{U}.$ Moreover, we also characterize the additive $\xi$-Lie derivation for $\xi\neq 1$ by its action at zero product in a unital algebra over $\mathbb{F}.$ UDC 512.5 Характеризацiя адитивного $\xi$ -диференцiювання Лi на унiтальних алгебрахНехай $\mathscr{R}$ --- комутативне кільце з одиницею, а $\mathscr{U}$~--- унітальна алгебра над $\mathscr{R}$ (або полем $\mathbb{F}$).$\mathscr{R}$-лінійне відображення $L:\mathscr{U}\rightarrow\mathscr{U}$ називається диференціюванням Лі на $\mathscr{U},$ якщо $L([u,v])=[L(u),v]+[u,L(v)]$ виконується для всіх $u,$ $v \in\mathscr{U}.$ Для скаляра $\xi\in\mathbb{F}$ адитивне відображення $L\colon \mathscr{U}\rightarrow\mathscr{U}$ називається адитивним $\xi$-диференціюванням Лі на $\mathscr{U},$ якщо $L([u,v]_{\xi})=[L(u),v]_{\xi}+[u,L(v)]_{\xi},$ де $[u,v]_{\xi}=uv-\xi vu$ виконується для всіх $u, v\in\mathscr{U}.$ У цій роботі при деяких припущеннях на $\mathscr{U}$ доведено, що кожне диференціювання Лі (відповідно, адитивне $\xi$-диференціювання Лі) ${L}$ на $\mathscr{U}$ має стандартний вигляд, тобто $L=\delta+\phi,$ де $\delta$ --- адитивне диференціювання на $\mathscr{U},$ а $\phi$ --- відображення $\phi\colon \mathscr{U}\rightarrow Z(\mathscr{U}),$ що зникає на $[u,v],$ якщо $uv=0$ у $\mathscr{U}.$ Більш того, охарактеризовано адитивне $\xi$-диференціювання Лі для $\xi\neq 1$ через його дію на нульовий добуток в унітальній алгебрі над $\mathbb{F}.$ Institute of Mathematics, NAS of Ukraine 2021-04-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/838 10.37863/umzh.v73i4.838 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 4 (2021); 455 - 466 Український математичний журнал; Том 73 № 4 (2021); 455 - 466 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/838/8999 |
| spellingShingle | Ashraf, M. Jabeen, A. Ashraf, M. Jabeen, A. Characterizations of additive $\xi$-Lie derivations on unital algebras |
| title | Characterizations of additive $\xi$-Lie derivations on unital algebras |
| title_alt | Characterizations of additive $\xi$-Lie derivations on unital algebras |
| title_full | Characterizations of additive $\xi$-Lie derivations on unital algebras |
| title_fullStr | Characterizations of additive $\xi$-Lie derivations on unital algebras |
| title_full_unstemmed | Characterizations of additive $\xi$-Lie derivations on unital algebras |
| title_short | Characterizations of additive $\xi$-Lie derivations on unital algebras |
| title_sort | characterizations of additive $\xi$-lie derivations on unital algebras |
| topic_facet | unital algebra Lie derivation ξ-Lie derivation unital algebra Lie derivation ξ-Lie derivation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/838 |
| work_keys_str_mv | AT ashrafm characterizationsofadditivexiliederivationsonunitalalgebras AT jabeena characterizationsofadditivexiliederivationsonunitalalgebras AT ashrafm characterizationsofadditivexiliederivationsonunitalalgebras AT jabeena characterizationsofadditivexiliederivationsonunitalalgebras |