Tri-additive maps and local generalized $(α,β)$-derivations
Let $R$ be a prime ring with nontrivial idempotents. We characterize a tri-additive map $f : R^3 \rightarrow R$ such that $f(x, y, z) = 0$ for all $x, y, z \in R$ with $xy = yz = 0$. As an application, we show that, in a prime ring with nontrivial idempotents, any local generalized $(\alpha , \bet...
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| author | Jamal, M. R. Mozumder, M. R. Ямал, М. Р. Мозамдер, М. Р. |
| author_facet | Jamal, M. R. Mozumder, M. R. Ямал, М. Р. Мозамдер, М. Р. |
| author_sort | Jamal, M. R. |
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| datestamp_date | 2019-12-05T09:25:15Z |
| description | Let $R$ be a prime ring with nontrivial idempotents. We characterize a tri-additive map $f : R^3 \rightarrow R$ such that $f(x, y, z) = 0$
for all $x, y, z \in R$ with $xy = yz = 0$. As an application, we show that, in a prime ring with nontrivial idempotents, any
local generalized $(\alpha , \beta)$-derivation (or a generalized Jordan triple $(\alpha , \beta)$-derivation) is a generalized $(\alpha , \beta)$-derivation. |
| first_indexed | 2026-03-24T02:11:44Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
M. R. Mozumder (Aligarh Muslim Univ., India), M. R. Jamal (Integral Univ., Lucknow, India)
TRI-ADDITIVE MAPS AND LOCAL GENERALIZED (\bfitalpha , \bfitbeta )-DERIVATIONS
ТРИАДИТИВНI ВIДОБРАЖЕННЯ
ТА ЛОКАЛЬНI УЗАГАЛЬНЕНI (\bfitalpha , \bfitbeta )-ПОХIДНI
Let R be a prime ring with nontrivial idempotents. We characterize a tri-additive map f : R3 \rightarrow R such that f(x, y, z) = 0
for all x, y, z \in R with xy = yz = 0. As an application, we show that, in a prime ring with nontrivial idempotents, any
local generalized (\alpha , \beta )-derivation (or a generalized Jordan triple (\alpha ,\beta )-derivation) is a generalized (\alpha , \beta )-derivation.
Нехай R — просте кiльце з нетривiальними iдемпотентами. Охарактеризовано триадитивне вiдображення f :
R3 \rightarrow R таке, що f(x, y, z) = 0 для всiх x, y, z \in R таких, що xy = yz = 0. Як застосування показано, що у
простому кiльцi з нетривiальними iдемпотентами довiльна локальна узагальнена (\alpha , \beta )-похiдна (або узагальнена
жорданова потрiйна (\alpha ,\beta )-похiдна) є узагальненою (\alpha , \beta )-похiдною.
1. Introduction. Throughout this paper, R denotes a prime ring with center Z(R), right (resp. left)
Martindale quotient ring Qr (resp. Q\ell ), and symmetric Martindale quotient ring Qs. Let Qmr (resp.
Qml ) denote the maximal right (resp. left) ring of quotients of R. We refer the reader to the book [1]
for details.
In [5], Chebotar, Ke and Lee characterized some maps preserving zero products: assume that
the ring R possesses nontrivial idempotents. If \phi : R \rightarrow R is a bijective additive map such that
\phi (x)\phi (y) = 0 whenever xy = 0, then \phi (xy)\phi (z) = \phi (x)\phi (yz) for any x, y, z \in R. Moreover,
if 1 \in R, then \phi (xy) = \lambda \phi (x)\phi (y) for any x, y \in R, where \lambda = \phi (1) - 1 \in C [5] (Theorem 3).
In [2], Brešar also discussed additive maps preserving zero products. In [6], Chuang and Lee
considered a general case, namely, a bi-additive map \phi : R\times R \rightarrow R such that \phi (x, y) = 0 whenever
xy = 0 (see Theorem 2.1). In this paper, we will generalize this result to a tri-additive map f :
R3 \rightarrow R such that f(x, y, z) = 0 whenever xy = yz = 0.
Let M be a R-bimodule. An additive mapping g : R \rightarrow M is called a generalized derivation
with associated derivation d : R \rightarrow M if g(xy) = g(x)y + xd(y) for all x, y \in R. In [11], Lee gave
a characterization of generalized derivations: every generalized derivation g on a dense right ideal
of R can be extended to Qmr and can be written in the form g(x) = ax+ d(x) for some a \in Qmr
and some derivation d on Qmr. Let \alpha , \beta : R \rightarrow R be automorphisms of R. An additive map \delta :
R \rightarrow M is called a skew derivation, or an (\alpha , \beta )-derivation, if \delta (xy) = \delta (x)\alpha (y) + \beta (x)\delta (y) for
any x, y \in R. An additive map g : R \rightarrow M is called a generalized (\alpha , \beta )-derivation if there is an
associated (\alpha , \beta )-derivation d : R \rightarrow M such that g(xy) = g(x)\alpha (y) + \beta (x)d(y) for any x, y \in R.
See [4] and [12] for a discussion of some of its properties.
An additive map d : R \rightarrow R is called a local derivation if for every x \in R there exists a
derivation dx : R \rightarrow R such that d(x) = dx(x). Kadison [8] and Larson and Sourour [9] asked under
what conditions a local derivation is a derivation. In [2], Brešar proved that a local derivation is a
derivation if R has nontrivial idempotents.
c\bigcirc M. R. MOZUMDER, M. R. JAMAL, 2017
848 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
TRI-ADDITIVE MAPS AND LOCAL GENERALIZED (\alpha , \beta )-DERIVATIONS 849
Recently, Wang generalized Brešar’s result to the case of generalized derivations. An additive
map g : R \rightarrow R is called a local generalized derivation if for every x \in R, there exists a generalized
derivation gx : R \rightarrow R such that g(x) = gx(x). Wang proved that a local generalized derivation is
actually a generalized derivation if R has nontrivial idempotents [14]. In Section 3, we will prove an
analogous result for generalized (\alpha , \beta )-derivations. Precisely, we will prove that a local generalized
(\alpha , \beta )-derivation on a prime ring with nontrivial idempotents is a generalized (\alpha , \beta )-derivation. We
will also prove that a generalized Jordan triple (\alpha , \beta )-derivation on a prime ring with nontrivial
idempotents is a generalized (\alpha , \beta )-derivation, which is a special case of [13] (Theorem 3).
2. Tri-additive maps preserving zero products. Let E be the additive subgroup generated by
all idempotents of R, and E denote the subring generated by E. Recall that in [5] Chebotar, Ke and
Lee proved that if \phi : R \rightarrow R is a bijective additive map such that \phi (x)\phi (y) = 0 whenever xy = 0,
then \phi (xy)\phi (z) = \phi (x)\phi (yz) for any x, y, z \in R. In [6], Chuang and Lee considered bi-additive
maps preserving zero products. We write their theorem in the following form.
Theorem 2.1 ([6], Theorem 2.3). Let R be a prime ring with nontrivial idempotents. Assume \phi :
R\times R \rightarrow R is a bi-additive map preserving zero products. Then there exists a nonzero ideal I such
that \phi (xa, y) = \phi (x, ay) for any x, y \in R and a \in I.
Note that because R has nontrivial idempotents, [E,E] \not = 0, and by examining the proof of
Theorem 2.1, we see that the nonzero ideal I can be chosen to be R[E,E]R. Moreover, R[E,E]R \subseteq
\subseteq E by Herstein’s arguments in [7, p. 4].
Now we consider a more general case. Let f : R3 \rightarrow R be a tri-additive map, that is, a map
f(x, y, z) that is is additive in each argument. In view of Theorem 2.1 and the proof in [6], we can
prove the following theorem.
Theorem 2.2. Let R be a prime ring with nontrivial idempotents. Let f(x, y, z) be a tri-additive
map with f(x, y, z) = 0 whenever xy = yz = 0. Then
f(xa, yb, z) - f(x, ayb, z) = f(xa, y, bz) - f(x, ay, bz) (2.1)
for all x, y, z \in R and a, b \in I, where I is some nonzero ideal of R.
Proof. For z \in R and e idempotent, define F (x, y)
df
= f(x, ye, (1 - e)z), then F (x, y) = 0 for
xy = 0. By Theorem 2.1 there exists a nonzero ideal I such that F (xa, y) = F (x, ay) for any a \in I.
That is,
f(xa, ye, (1 - e)z) = f(x, aye, (1 - e)z). (2.2)
Note that by the remark after Theorem 2.1, the choice of I is independent of e and z. In fact, we
can choose I = R[E,E]R. Thus, (2.2) holds for any x, y, z \in R, any a \in I and any idempotent e.
Analogously,
f(xa, y(1 - e), ez) = f(x, ay(1 - e), ez). (2.3)
Comparing (2.2) and (2.3), we see that
f(xa, ye, z) - f(x, aye, z) = f(xa, y, ez) - f(x, ay, ez).
It can be easily checked that
f(xa, ye, z) - f(x, aye, z) = f(xa, y, ez) - f(x, ay, ez)
for any x, y, z \in R, any a \in I, and any e \in E. Because I = R[E,E]R \subseteq E, we get
f(xa, yb, z) - f(x, ayb, z) = f(xa, y, bz) - f(x, ay, bz)
for any x, y, z \in R, any a, b \in I, as asserted.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
850 M. R. MOZUMDER, M. R. JAMAL
3. Generalized (\bfitalpha , \bfitbeta )-derivations. Let \alpha , \beta be automorphisms of R, and let M be an
R-bimodule. Recall that an additive map g : R \rightarrow M is a generalized (\alpha , \beta )-derivation if g(xy) =
= g(x)\alpha (y) + \beta (x)d(y) for some (\alpha , \beta )-derivation d : R \rightarrow M.
Here we need a property on extensions of (\alpha , \beta )-derivations. It is well known that the automor-
phisms of R and (\alpha , \beta )-derivations of R can be uniquely extended to Qm\ell . We want to show that an
(\alpha , \beta )-derivation from a nonzero ideal to Qm\ell can be also extended to an (\alpha , \beta )-derivation of Qm\ell .
The proof simply follows the standard arguments in [10] (Lemma 2) and [11] (Theorem 2) for the
case of derivations. For brevity, we only sketch it here.
Proposition 3.1. Let R be a prime ring, I be a nonzero ideal of R, and \alpha , \beta be automorphisms
of R. Then every (\alpha , \beta )-derivation \delta : R \rightarrow Qm\ell can be uniquely extended to an (\alpha , \beta )-derivation \~\delta :
Qm\ell \rightarrow Qm\ell . Moreover, every (\alpha , \beta )-derivation \delta : I \rightarrow Q\ell can be uniquely extended to an (\alpha , \beta )-
derivation \~\delta : Qm\ell \rightarrow Qm\ell .
Proof (Sketch of Proof). Let \delta : R \rightarrow Qm\ell be an (\alpha , \beta )-derivation. For any q \in Qm\ell
choose a dense left ideal \lambda of R such that \lambda q \subseteq R. Define \phi : Qm\ell \lambda \rightarrow Qm\ell by \phi
\Bigl( \sum
uiai
\Bigr)
=
=
\sum
ui\beta
- 1
\bigl(
(\delta (aiq) - \delta (ai)\alpha (q))
\bigr)
, where ui \in Qm\ell and ai \in \lambda . Then \phi is a right multiplier
induced by an element \^q in the maximal left quotient ring of Qm\ell , which is just Qm\ell itself (see
Proposition 2.1.7 and Theorem 2.1.11 in [1]). In this sense, \delta can be extended to a map \~\delta :
Qm\ell \rightarrow Qm\ell by defining \~\delta (q)
df
=\beta (\^q). It can be checked that \~\delta is an (\alpha , \beta )-derivation of Qm\ell and
that this extension is unique. The second part of the proof simply follows the arguments in [11]
(Theorem 2).
Now we can prove the following theorem.
Theorem 3.1. Let R be a prime ring with nontrivial idempotents. If g : R \rightarrow R is an additive
map such that \beta (x)g(y)\alpha (z) = 0 for any x, y, z \in R with xy = yz = 0, then g is a generalized
(\alpha , \beta )-derivation.
Proof. Because R possesses nontrivial idempotents, by Theorem 2.2 we know that
\beta (xa)g(yb)\alpha (z) - \beta (x)g(ayb)\alpha (z) = \beta (xa)g(y)\alpha (bz) - \beta (x)g(ay)\alpha (bz) (3.1)
for any x, y, z \in R and a, b \in I, where I is a nonzero ideal of R. Because R is prime and \alpha , \beta are
automorphisms and rearranging the terms, the equation (3.1) can be reduced to
\beta (a)(g(yb) - g(y)\alpha (b)) = g(ayb) - g(ay)\alpha (b). (3.2)
Now, define Fb(y) = g(yb) - g(y)\alpha (b); then (3.2) becomes \beta - 1(Fb(ay)) = a\beta - 1(Fb(y)). That is,
\beta - 1Fb is a left I-module map and hence a left R-module map. Therefore, \beta - 1Fb is a right multiplier
induced by an element in Q\ell (see [1], Proposition 2.2.1). This implies
g(yb) - g(y)\alpha (b) = \beta (y)d(b), (3.3)
for any y \in R and any b \in I, where d : I \rightarrow Q\ell is an additive map. For y \in R and b, c \in I, by (3.3)
g(ybc) - g(y)\alpha (bc) = \beta (y)d(bc). (3.4)
Expanding otherwise and simplifying, the equation (3.4) reduces to
g(ybc) - (g(y)\alpha (b) + \beta (y)d(b))\alpha (c) = \beta (yb)d(c). (3.5)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
TRI-ADDITIVE MAPS AND LOCAL GENERALIZED (\alpha , \beta )-DERIVATIONS 851
Combining (3.4) and (3.5), we obtain d(bc) = d(b)\alpha (c) + \beta (b)d(c), so d : I \rightarrow Q\ell is an (\alpha , \beta )-
derivation. Up to now we have
g(xa) = g(x)\alpha (a) + \beta (x)d(a) (3.6)
for any x \in R and any a \in I, where d : I \rightarrow Q\ell is an (\alpha , \beta )-derivation. By Proposition 3.1, d can
be uniquely extended to an (\alpha , \beta )-derivation of Q\ell , which we still denote by d. For any x, y \in R
and any a \in I, by (3.6)
g(x(ya)) = g(x)\alpha (ya) + \beta (x)d(ya) =
= g(x)\alpha (y)\alpha (a) + \beta (x)d(y)\alpha (a) + \beta (x)\beta (y)d(a). (3.7)
On the other hand,
g((xy)a) = g(xy)\alpha (a) + \beta (xy)d(a). (3.8)
Comparing (3.7), (3.8) and using the primeness, we get g(xy) = g(x)\alpha (y) + \beta (x)d(y) for any
x, y \in R. This means that g is a generalized (\alpha , \beta )-derivation of R.
By analogy with the local derivations and local generalized derivations mentioned in Section 1,
we introduce the following notion.
Definition 3.1. An additive map g : R \rightarrow R is called a local generalized (\alpha , \beta )-derivation if
for every x \in R, there exists a generalized (\alpha , \beta )-derivation gx, which depends on x, such that
g(x) = gx(x).
The following theorem shows that a local generalized (\alpha , \beta )-derivation is a generalized (\alpha , \beta )-
derivation. This generalizes the derivation case in [2] and the generalized derivation case in [14].
Theorem 3.2. Let R be a prime ring with nontrivial idempotents, and let \alpha , \beta be automorphisms
of R. Then a local generalized (\alpha , \beta )-derivation is a generalized (\alpha , \beta )-derivation.
Proof. Let g be a local generalized (\alpha , \beta )-derivation of R. For every y \in R, there is a
generalized (\alpha , \beta )-derivation gy with associated (\alpha , \beta )-derivation dy such that g(y) = gy(y). Hence
for any x, y, z \in R with xy = yz = 0, we have
\beta (x)g(y)\alpha (z) = \beta (x)gy(y)\alpha (z) = \beta (x)gy(yz) - \beta (xy)dy(z) = 0.
By Theorem 3.1, g is actually a generalized (\alpha , \beta )-derivation.
Recall that an additive map \delta : R \rightarrow R is called a Jordan triple (\alpha , \beta )-derivation, if
\delta (xyx) = \delta (x)\alpha (y)\alpha (x) + \beta (x)\delta (y)\alpha (x) + \beta (x)\beta (y)\delta (x) (3.9)
for any x, y \in R. An additive map g : R \rightarrow R is called a generalized Jordan triple (\alpha , \beta )-derivation
if there exists a Jordan triple (\alpha , \beta )-derivation \delta of R such that
g(xyx) = g(x)\alpha (y)\alpha (x) + \beta (x)\delta (y)\alpha (x) + \beta (x)\beta (y)\delta (x) (3.10)
for any x, y \in R.
In [13], Liu and Shiue proved that a generalized Jordan triple (\alpha , \beta )-derivation on a 2-torsion free
semiprime ring must be a generalized (\alpha , \beta )-derivation [13] (Theorem 3). Now we want to prove
an analogous theorem for the special case of prime rings with nontrivial idempotents, but where the
associated map \delta in (3.10) is any map.
In order to prove the theorem, we need a result in functional identities.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
852 M. R. MOZUMDER, M. R. JAMAL
Lemma 3.1. Let R be a prime ring and \alpha , \beta : R \rightarrow R be automorphisms of R. If F,G :
R \rightarrow R are two additive maps such that F (x)\alpha (y) = \beta (x)G(y) for any x, y \in R, then there exists
an element q \in Qs such that F (x) = \beta (x)q and G(y) = q\alpha (y).
Proof. It is well known that any automorphism of R can be uniquely extended to an automor-
phism of Qs, Q\ell , or Qr. A direct computation shows that F (rx)\alpha (y) - \beta (r)F (x)\alpha (y) = 0 for
any r, x, y \in R, so because R is prime, we see that F (rx) = \beta (r)F (x). That is, \beta - 1F is a left
R-module map of R. Therefore, there exists an element s \in Q\ell such that \beta - 1F (x) = xs. Hence,
F (x) = \beta (x)q, where q = \beta (s) \in Q\ell . By assumption we have \beta (x)q\alpha (y) = \beta (x)G(y), which
implies that G(y) = q\alpha (y) because R is a prime ring. Moreover, q is an element of Qs because
qR \subseteq R.
Theorem 3.3. Let R be a prime ring with nontrivial idempotents, and let \alpha , \beta be automorphisms
of R. If g : R \rightarrow R is an additive map and d : R \rightarrow R is any map such that
g(xyx) = g(x)\alpha (y)\alpha (x) + \beta (x)d(y)\alpha (x) + \beta (x)\beta (y)d(x) (3.11)
for any x, y \in R, then g is a generalized (\alpha , \beta )-derivation with the associated derivation \delta , and one
of the following holds:
(1) d = \delta , is exactly the associated (\alpha , \beta )-derivation of g;
(2) char R = 2 and there exists an invertible element q \in Qs, such that d(x) = \delta (x) + \beta (x)q =
= \delta (x) - q\alpha (x) and \beta (x) = q\alpha (x)q - 1.
Proof. For any s \in R and x, y, z \in R with xy = yz = 0, it follows from (3.11) that
0 = \beta (x)g(yzsy) = \beta (x)g(y)\alpha (z)\alpha (s)\alpha (y).
Because \alpha , \beta are automorphisms and R is prime, we have \beta (x)g(y)\alpha (z) = 0 or \alpha (y) = 0. Take
I1 = \{ y \in R | \beta (x)g(y)\alpha (z) = 0\} for all x, z \in R and I2 = \{ y \in R | \alpha (y) = 0\} . Clearly,
I1 and I2 both are additive subgroups of R, whose union is R. But, a group can not be union of
two of its proper subgroups. Hence, either I1 = R and I2 = R. But, if I2 = R gives \alpha = 0,
a contradiction. Hence, \beta (x)g(y)\alpha (z) = 0 for all x, y, z \in R with xy = yz = 0. Hence g is a
generalized (\alpha , \beta )-derivation with associated (\alpha , \beta )-derivation \delta by Theorem 3.1.
Now we claim that d is additive. Substituting y by y + z in (3.11), and because g, \alpha and \beta are
all additive, we get
\beta (x)
\bigl(
d(y + z) - d(y) - d(z)
\bigr)
\alpha (x) = 0. (3.12)
Linearizing on x, it follows that
\beta (u)
\bigl(
d(y + z) - d(y) - d(z)
\bigr)
\alpha (x) + \beta (x)
\bigl(
d(y + z) - d(y) - d(z)
\bigr)
\alpha (u) = 0. (3.13)
Substituting u by ux in (3.13) and using (3.12), we see that
\beta (x)
\bigl(
d(y + z) - d(y) - d(z)
\bigr)
\alpha (ux) = 0
for all u, x, y, z \in R. Again, because \alpha is an automorphism and R is prime, \beta (x)
\bigl(
d(y+ z) - d(y) -
- d(z)
\bigr)
= 0 or \alpha (x) = 0 for all x, y, z \in R. As discuss in the beginning of the theorem, we have
\beta (x)
\bigl(
d(y + z) - d(y) - d(z)
\bigr)
= 0 for all x, y, z \in R. This implies that d(y + z) = d(y) + d(z) for
all y, z \in R. That is, d is additive.
Now g is a generalized (\alpha , \beta )-derivation with associated (\alpha , \beta )-derivation \delta . From (3.9) and
(3.11) we get
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
TRI-ADDITIVE MAPS AND LOCAL GENERALIZED (\alpha , \beta )-DERIVATIONS 853
\beta (x)d(y)\alpha (x) + \beta (x)\beta (y)d(x) = \beta (x)\delta (y)\alpha (x) + \beta (x)\beta (y)\delta (x)
for all x, y \in R, and hence d(y)\alpha (x)+\beta (y)d(x) = \delta (y)\alpha (x)+\beta (y)\delta (x). That is, (d - \delta )(y)\alpha (x)+
+ \beta (y)(d - \delta )(x) = 0. Because d - \delta is additive, it follows by Lemma 3.1 that (d - \delta )(x) =
= \beta (x)q = - q\alpha (x) for some q \in Qs, which means that d(x) = \delta (x) + \beta (x)q = \delta (x) - q\alpha (x). For
any x, y \in R, we have
\beta (xy)q = \beta (x)\beta (y)q = - \beta (x)q\alpha (y) = q\alpha (x)\alpha (y).
Therefore, 2qR2 = 0, and this implies that 2q = 0. If char R \not = 2, then q = 0 and d = \delta , as asserted.
In case char R = 2 and q \not = 0, by \beta (x)q = - q\alpha (x) = q\alpha (x) we can conclude that q is invertible in
Qs and hence \beta (x) = q\alpha (x)q - 1.
The following is a special case of [3] (Theorem 1).
Corollary 3.1. Let R be a prime ring with nontrivial idempotents and \alpha , \beta be automorphisms of
R. If \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r} (R) \not = 2 and d : R \rightarrow R is a Jordan triple (\alpha , \beta )-derivation, then d is an (\alpha , \beta )-derivation.
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Received 11.04.13,
after revision — 19.02.17
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 6
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| id | umjimathkievua-article-1739 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:44Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/46/a0937bbd4cb13cafbb507f7da602c746.pdf |
| spelling | umjimathkievua-article-17392019-12-05T09:25:15Z Tri-additive maps and local generalized $(α,β)$-derivations Триадитивнi вiдображення та локальнi узагальненi $(α,β)$-похiднi Jamal, M. R. Mozumder, M. R. Ямал, М. Р. Мозамдер, М. Р. Let $R$ be a prime ring with nontrivial idempotents. We characterize a tri-additive map $f : R^3 \rightarrow R$ such that $f(x, y, z) = 0$ for all $x, y, z \in R$ with $xy = yz = 0$. As an application, we show that, in a prime ring with nontrivial idempotents, any local generalized $(\alpha , \beta)$-derivation (or a generalized Jordan triple $(\alpha , \beta)$-derivation) is a generalized $(\alpha , \beta)$-derivation. Нехай $R$ — просте кiльце з нетривiальними iдемпотентами. Охарактеризовано триадитивне вiдображення $f : R^3 \rightarrow R$ таке, що $f(x, y, z) = 0$ для всiх $x, y, z \in R$ таких, що $xy = yz = 0$. Як застосування показано, що у простому кiльцi з нетривiальними iдемпотентами довiльна локальна узагальнена $(\alpha , \beta )$-похiдна (або узагальнена жорданова потрiйна $(\alpha , \beta )$-похiдна) є узагальненою $(\alpha , \beta)$-похiдною. Institute of Mathematics, NAS of Ukraine 2017-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1739 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 6 (2017); 848-853 Український математичний журнал; Том 69 № 6 (2017); 848-853 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1739/721 Copyright (c) 2017 Jamal M. R.; Mozumder M. R. |
| spellingShingle | Jamal, M. R. Mozumder, M. R. Ямал, М. Р. Мозамдер, М. Р. Tri-additive maps and local generalized $(α,β)$-derivations |
| title | Tri-additive maps and local generalized $(α,β)$-derivations |
| title_alt | Триадитивнi вiдображення та локальнi узагальненi $(α,β)$-похiднi |
| title_full | Tri-additive maps and local generalized $(α,β)$-derivations |
| title_fullStr | Tri-additive maps and local generalized $(α,β)$-derivations |
| title_full_unstemmed | Tri-additive maps and local generalized $(α,β)$-derivations |
| title_short | Tri-additive maps and local generalized $(α,β)$-derivations |
| title_sort | tri-additive maps and local generalized $(α,β)$-derivations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1739 |
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