Some algebraic identities in 3-prime near-rings
We extend the domain of applicability of the concept of $(1, \alpha)$-derivations in $3$-prime near-rings by analyzing the structure and commutativity of near-rings admitting $(1, \alpha)$-derivations satisfying certain differential identities.
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| author | Boua, A. Ashraf, M. Boua, A. Ashraf, M. Boua, A. Ashraf, M. |
| author_facet | Boua, A. Ashraf, M. Boua, A. Ashraf, M. Boua, A. Ashraf, M. |
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| description | We extend the domain of applicability of the concept of $(1, \alpha)$-derivations in $3$-prime near-rings by analyzing the structure and commutativity of near-rings admitting $(1, \alpha)$-derivations satisfying certain differential identities. |
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UDC 512.5
A. Boua (Sidi Mohamed Ben Abdellah Univ., Fez, Morocco),
M. Ashraf (Aligarh Muslim Univ., India)
SOME ALGEBRAIC IDENTITIES IN \bfthree -PRIME NEAR-RINGS
ДЕЯКI АЛГЕБРАЇЧНI ТОТОЖНОСТI ДЛЯ \bfthree -ПРОСТИХ МАЙЖЕ КIЛЕЦЬ
We extend the domain of applicability of the concept of (1, \alpha )-derivations in 3-prime near-rings by analyzing the structure
and commutativity of near-rings admitting (1, \alpha )-derivations satisfying certain differential identities.
Розширено область застосовностi поняття (1, \alpha )-похiдних для 3-простих майже кiлець, як результат вивчення
структури та комутативностi майже кiлець, що допускають (1, \alpha )-похiднi, якi задовольняють деякi диференцiальнi
тотожностi.
1. Introduction. Throughout this paper, \scrN will denote a zero-symmetric left near-ring. A near-ring
\scrN is called zero symmetric if 0x = 0 for all x \in \scrN (recall that in a left near ring x0 = 0 for all
x \in \scrN ). \scrN is called 3-prime if x\scrN y = \{ 0\} implies x = 0 or y = 0. The symbol Z(\scrN ) will
represent the multiplicative center of \scrN , that is, Z(\scrN ) = \{ x \in \scrN | xy = yx for all y \in \scrN \} . For
any x, y \in \scrN , as usual, [x, y] = xy - yx and x \circ y = xy + yx will denote the well-known Lie
product and Jordan product, respectively. Recall that \scrN is called 2-torsion free if 2x = 0 implies
x = 0 for all x \in \scrN . For terminologies concerning near-rings we refer to G. Pilz [6].
An additive mapping d : \scrN \rightarrow \scrN is said to be a derivation if d(xy) = xd(y) + d(x)y for all
x, y \in \scrN , or, equivalently, as noted in [7], that d(xy) = d(x)y + xd(y) for all x, y \in \scrN . An
additive mapping d : \scrN \rightarrow \scrN is called a semiderivation if there exists a function g : \scrN \rightarrow \scrN such
that d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and d(g(x)) = g(d(x)) hold for all x, y \in \scrN .
An additive mapping d : \scrN \rightarrow \scrN is called a two sided \alpha -derivation if there exists a function \alpha :
\scrN \rightarrow \scrN such that d(xy) = d(x)y+\alpha (x)d(y) and d(xy) = d(x)\alpha (y)+xd(y) hold for all x, y \in \scrN .
An additive mapping d : \scrN \rightarrow \scrN is called a (1, \alpha )-derivation if there exists a function \alpha : \scrN \rightarrow \scrN
such that d(xy) = d(x)y + \alpha (x)d(y) holds for all x, y \in \scrN . An additive mapping d : \scrN \rightarrow \scrN is
called an (\alpha , 1)-derivation if there exists a function \alpha : \scrN \rightarrow \scrN such that d(xy) = d(x)\alpha (y)+xd(y)
holds for all x, y \in \scrN . Obviously, a two sided \alpha -derivation is both (1, \alpha )-derivation as well
as (\alpha , 1)-derivation. Also, any derivation on \scrN is a (1, \alpha )-derivation, but the converse is not
true in general (see [5]). There are several results asserting that 3-prime near-rings with certain
constrained derivations have ringlike behavior. Recently many authors (see [1, 2, 4], where further
references can be found) studied commutativity of 3-prime near-rings satisfying certain identities
involving derivations, semiderivations and two sided \alpha -derivations. Now our aim is to study the
commutativity behavior of a 3-prime near-ring which admits (1, \alpha )-derivations satisfying certain
properties. In fact, our results generalize, extend and complement several results obtained earlier in
[1, 5, 8] on derivations, semiderivations and two sided \alpha -derivations for 3-prime near-rings.
2. Some preliminaries. In this section, we include some well-known results which will be used
for developing the proof of our main result.
Lemma 2.1 ([4], Theorem 2.9). Let \scrN be a 3-prime near-ring. If I is a nonzero semigroup
ideal of \scrN and d is a nonzero derivation of \scrN , then the following assertions are equivalent:
(i) [u, v] \in Z(\scrN ) for all u, v \in I,
c\bigcirc A. BOUA, M. ASHRAF, 2020
36 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
SOME ALGEBRAIC IDENTITIES IN 3-PRIME NEAR-RINGS 37
(ii) [d(u), v] \in Z(\scrN ) for all u, v \in I,
(iii) \scrN is a commutative ring.
Lemma 2.2 ([4], Theorem 2.10). Let \scrN be a 2-torsion free 3-prime near-ring. If u\circ v \in Z(\scrN )
for all u, v \in \scrN , then \scrN is a commutative ring.
Lemma 2.3 ([3], Lemma 1.5). Let \scrN be a 3-prime near-ring. If \scrN \subseteq Z(\scrN ), then \scrN is a
commutative ring.
Lemma 2.4. A near-ring \scrN admits a (1, \alpha )-derivation d associated with an additive map \alpha if
and only if it is zero-symmetric.
Proof. Let \scrN be a zero-symmetric near-ring. Then the zero map is a (1, \alpha )-derivation d on \scrN .
Conversely, assume that \scrN has an (1, \alpha )-derivation d associated with an additive map \alpha . Let x, y
be two arbitrary elements of \scrN . By definition of d, we have
d(x0y) = d(x(0y)) = d(x)(0y) + \alpha (x)d(0y) =
= (d(x)0)y + \alpha (x)(d(0)y) + \alpha (x)(\alpha (0)d(y)) = 0y + (\alpha (x)d(0))y + (\alpha (x)\alpha (0))d(y) =
= 0y + (\alpha (x)0)y + (\alpha (x)\alpha (0))d(y) = 0y + 0y + (\alpha (x)0)d(y) = 0y + 0y + 0d(y).
On the other hand,
d(x0y) = d((x0)y)) = d(0y) =
= d(0)y + \alpha (0)d(y) = 0y + 0d(y).
By comparing the last two expressions, we find that 0y = 0 for all y \in \scrN , and hence \scrN is a
zero-symmetric left near-ring.
Remark. The above lemma has its independent interest in the study of arbitrary left near-rings
(not necessarily zero-symmetric). It can also be easily seen that it is also true in the case of right
near-ring.
Lemma 2.5. Let \scrN be a near-ring and d be a (1, \alpha )-derivation associated with a map \alpha . Then
\scrN satisfies the following property:
(d(x)y + \alpha (x)d(y)) z =
= d(x)yz + \alpha (x)d(y)z + \alpha (x)\alpha (y)d(z) - \alpha (xy)d(z) for all x, y, z \in \scrN .
Proof. From the associative law we have
d((xy)z) = d(xy)z + \alpha (xy)d(z) =
= (d(x)y + \alpha (x)d(y))z + \alpha (xy)d(z) for all x, y \in \scrN .
Also
d(x(yz)) = d(x)yz + \alpha (x)d(yz) =
= d(x)yz + \alpha (x)d(y)z + \alpha (x)\alpha (y)d(z) for all x, y, z \in \scrN .
Combining the above two equalities, we find
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
38 A. BOUA, M. ASHRAF
(d(x)y + \alpha (x)d(y)) z + \alpha (xy)d(z) =
= d(x)yz + \alpha (x)d(y)z + \alpha (x)\alpha (y)d(z) for all x, y, z \in \scrN ,
which is the required result.
Lemma 2.6. Let \scrN be a 3-prime near-ring and d be a nonzero (1, \alpha )-derivation associated
with an onto map \alpha .
(i) If ad(\scrN ) = \{ 0\} , a \in \scrN and \alpha is an onto map, then a = 0.
(ii) If d(\scrN )a = \{ 0\} and \alpha (xy) = \alpha (x)\alpha (y) for all x, y \in \scrN , then a = 0.
Proof. (i) If ad(\scrN ) = \{ 0\} and a \in \scrN , then ad(xy) = 0 for all x, y \in \scrN . This implies that
ad(x)y + a\alpha (x)d(y) = 0 for all x, y \in \scrN , and, hence, a\alpha (x)d(y) = 0 for all x, y \in \scrN . Since \alpha
is onto, a\scrN d(y) = \{ 0\} for all y \in \scrN . By 3-primeness of \scrN and d \not = 0, we obtain a = 0.
(ii) If d(\scrN )a = \{ 0\} , then d(xy)a = 0 for all x, y \in \scrN . By Lemma 2.5, we get d(x)ya +
+ \alpha (x)d(y)a+ \alpha (x)\alpha (y)d(a) - \alpha (xy)d(a) = 0 for all x, y \in \scrN . By the given hypothesis, we find
that d(x)ya = 0 for all x, y \in \scrN , i.e., d(x)\scrN a = \{ 0\} for all x \in \scrN . Since d \not = 0 and \scrN is
3-prime, we arrive at a = 0.
Lemma 2.7. Let \scrN be a 2-torsion free 3-prime near-ring. If d is a nonzero (1, \alpha )-derivation
associated with an onto map \alpha such that \alpha d = d\alpha , then d2 \not = 0.
Proof. Suppose that d2(\scrN ) = \{ 0\} . Then, for x, y \in \scrN , one can write
0 = d2(xy) = d(d(xy)) = d(d(x)y + \alpha (x)d(y)) =
= d2(x)y + \alpha (d(x))d(y) + d(\alpha (x))d(y) + \alpha 2(x)d2(y) =
= \alpha (d(x))d(y) + d(\alpha (x))d(y) for all x, y \in \scrN .
Note that \alpha (d(x)) = d(\alpha (x)), we find that
2\alpha (d(x))d(y) = 0 for all x, y \in \scrN .
Since \scrN is 2-torsion free, we arrive at
d(\alpha (x))d(y) = 0 for all x, y \in \scrN .
By using Lemma 2.6 and the fact that \alpha is onto, we obtain that d = 0 a contradiction.
3. Main results. In [2], H. E. Bell and G. Mason proved that a 3-prime near-ring \scrN must be
commutative if it admits a derivation d such that d(\scrN ) \subseteq Z(\scrN ). This result was generalized by
the authors in [5, 8]. They replaced the derivation with a semiderivation or two sided \alpha -derivation.
Our objective in the following theorems is to generalize these results by treating the case of (1, \alpha )-
derivation where \alpha is an onto map.
Theorem 3.1. Let \scrN be a 2-torsion free 3-prime near-ring. If \scrN admits a nonzero (1, \alpha )-
derivation d associated with an onto map \alpha such that \alpha d = d\alpha and d(\scrN ) \subseteq Z(\scrN ), then \scrN is a
commutative ring.
Proof. Suppose that d(\scrN ) \subseteq Z(\scrN ). By definition of d, we have
(d(x)y + \alpha (x)d(y))z = z(d(x)y + \alpha (x)d(y)) for all x, y, z \in \scrN .
This implies that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
SOME ALGEBRAIC IDENTITIES IN 3-PRIME NEAR-RINGS 39
d(x)yz + \alpha (x)d(y)z + \alpha (x)\alpha (y)d(z) - \alpha (xy)d(z) =
= zd(x)y + z\alpha (x)d(y) for all x, y, z \in \scrN . (3.1)
Replacing z by d(z) in (3.1), we get
\alpha (x)\alpha (y)d2(z) = \alpha (xy)d2(z) for all x, y, z \in \scrN ,
which reduces to
d2(z)\scrN
\bigl(
\alpha (x)\alpha (y) - \alpha (xy)
\bigr)
= \{ 0\} for all x, y, z \in \scrN . (3.2)
In view of 3-primeness of \scrN , (3.2) implies that
d2 = 0 or \alpha (xy) = \alpha (x)\alpha (y) for all x, y \in \scrN .
Since d \not = 0, we obtain d2 \not = 0 by Lemma 2.7, and in this case the previous relation becomes only
\alpha (xy) = \alpha (x)\alpha (y) for all x, y \in \scrN and by (3.1) we arrive at
d(x)yz + \alpha (x)d(y)z = zd(x)y + z\alpha (x)d(y) for all x, y, z \in \scrN . (3.3)
Replacing z by \alpha (x) in (3.3), we obtain
d(x)y\alpha (x) = \alpha (x)d(x)y for all x, y \in \scrN .
This yields that
d(x)\scrN [\alpha (x), y] = \{ 0\} for all x, y \in \scrN .
Since \scrN is 3-prime, we find that
d(x) = 0 or \alpha (x) \in Z(\scrN ) for all x \in \scrN . (3.4)
Suppose there exists x0 \in \scrN such that d(x0) = 0. Replacing x by x0 in (3.1), we get \alpha (x0)d(y)z =
= z\alpha (x0)d(y) for all y, z \in \scrN , which implies that
d(y)\scrN [\alpha (x0), z] = \{ 0\} for all y, z \in \scrN .
Since \scrN is 3-prime and d \not = 0, the last expression implies that \alpha (x0) \in Z(\scrN ), and the relation (3.4)
yields \alpha (x) \in Z(\scrN ) for all x \in \scrN . Now in this case (3.1) becomes
d(x)\scrN [y, z] = \{ 0\} for all y, z \in \scrN .
Since \scrN is 3-prime and d \not = 0, we conclude that \scrN \subseteq Z(\scrN ) and by Lemma 2.3, \scrN is a commutative
ring.
Corollary 3.1 ([2], Theorem 2). Let \scrN be a 2-torsion free 3-prime near-ring. If \scrN admits a
nonzero derivation d such that d(\scrN ) \subseteq Z(\scrN ), then \scrN is a commutative ring.
Corollary 3.2. Let \scrN be a 2-torsion free 3-prime near-ring. If \scrN admits a nonzero semideriva-
tion d associated with an onto map \alpha such that d(\scrN ) \subseteq Z(\scrN ), then \scrN is a commutative ring.
Theorem 3.2. Let \scrN be a 2-torsion free 3-prime near-ring which admits a nonzero (1, \alpha )-
derivation d associated with an onto map \alpha . Then the following assertions are equivalent:
(i) d([x, y]) = 0 for all x, y \in \scrN ,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
40 A. BOUA, M. ASHRAF
(ii) \scrN is a commutative ring.
Proof. It is easy to verify that (ii) \Rightarrow (i).
(i) \Rightarrow (ii) Suppose that d([x, y]) = 0 for all x, y \in \scrN . Replacing y by xy, we get
0 = d([x, xy]) = d(x[x, y]) =
= d(x)[x, y] + \alpha (x)d([x, y]) for all x, y \in \scrN .
This implies that
d(x)xy = d(x)yx for all x, y \in \scrN . (3.5)
Replacing y by yt in (3.5) and using it again, we get
d(x)ytx = d(x)xyt =
= d(x)yxt for all x, y, t \in \scrN ,
which reduces to
d(x)\scrN [x, t] = \{ 0\} for all x, t \in \scrN .
By 3-primeness of \scrN , we obtain
d(x) = 0 or x \in Z(\scrN ) for all x \in \scrN . (3.6)
Suppose there exists x0 \in \scrN such that d(x0) = 0. Then by hypothesis, we have d(x0y) = d(yx0)
for all y \in \scrN . Since \scrN is zero-symmetric by Lemma 2.5, the last equation implies that
\alpha (x0)d(y) = d(y)x0 for all y \in \scrN . (3.7)
Taking yt instead of y in (3.7) and using Lemma 2.5 together with the fact that d(x0) = 0, we find
that
\alpha (x0)d(y)t+ \alpha (x0)\alpha (y)d(t) = d(y)tx0 + \alpha (y)d(t)x0 for all y, t \in \scrN .
By (3.7) the above expression implies that
d(y)x0t+ \alpha (x0)\alpha (y)d(t) = d(y)tx0 + \alpha (y)\alpha (x0)d(t) for all y, t \in \scrN .
Putting [u, v] instead of t in the last expression, we get
d(y)x0[u, v] = d(y)[u, v]x0 for all u, v, y \in \scrN . (3.8)
Replacing y by yt in (3.8) and using it again, we obtain
d(y)tx0[u, v] + \alpha (y)d(t)x0[u, v] =
= d(y)t[u, v]x0 + \alpha (y)d(t)[u, v]x0 for all y, t \in \scrN ,
which reduces to
d(y)\scrN
\bigl(
x0[u, v] - [u, v]x0
\bigr)
= \{ 0\} for all y, u, v \in \scrN .
Since d \not = 0, by 3-primeness of \scrN , we get x0[u, v] - [u, v]x0 = 0 for all u, v \in \scrN in this case (3.6)
becomes [u, v] \in Z(\scrN ) for all u, v \in \scrN and by Lemma 2.1, we conclude that \scrN is a commutative
ring.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
SOME ALGEBRAIC IDENTITIES IN 3-PRIME NEAR-RINGS 41
Corollary 3.3 ([1], Theorem 4.1). Let \scrN be a 2-torsion free 3-prime near-ring. If \scrN admits a
nonzero derivation d such that d([x, y]) = 0 for all x, y \in \scrN , then \scrN is a commutative ring.
Corollary 3.4. Let \scrN be a 2-torsion free 3-prime near-ring. If \scrN admits a nonzero semideriva-
tion d associated with an onto map \alpha such that d([x, y]) = 0 for all x, y \in \scrN , then \scrN is a
commutative ring.
Theorem 3.3. Let \scrN be a 2-torsion free 3-prime near-ring. Then there exists no nonzero
(1, \alpha )-derivation d associated with an onto map \alpha such that d(x \circ y) = 0 for all x, y \in \scrN .
Proof. Assume that d(x \circ y) = 0 for all x, y \in \scrN . Replacing y by xy, we get
0 = d(x \circ xy) = d(x(x \circ y)) =
= d(x)(x \circ y) + \alpha (x)d(x \circ y) for all x, y \in \scrN .
This implies that
d(x)xy = - d(x)yx for all x, y \in \scrN . (3.9)
Replacing y by yt in (3.9) and using it again, we get
d(x)ytx = - d(x)xyt = d(x)xy( - t) =
= ( - d(x)yx)( - t) = d(x)y( - x)( - t) for all x, y, t \in \scrN .
This can be rewritten as
d(x)\scrN ( - t( - x) + ( - x)t) = \{ 0\} for all x, y, t \in \scrN .
By 3-primeness of \scrN , the latter equation becomes
d(x) = 0 or - x \in Z(\scrN ) for all x \in \scrN . (3.10)
Suppose there exists x0 \in \scrN such that d(x0) = 0. Then by hypothesis, we have d(x0y) = - d(yx0)
for all y \in \scrN . Since \scrN is zero-symmetric, by definition of d the last equation implies that
\alpha (x0)d(y) = - d(y)x0 for all y \in \scrN . (3.11)
Taking yt instead of y in (3.11) and using Lemma 2.5 together with the fact that d(x0) = 0, we find
that
\alpha (x0)d(y)t+ \alpha (x0)\alpha (y)d(t) = - \alpha (y)d(t)x0 - d(y)tx0 for all y, t \in \scrN .
By (3.11) the above expression implies that
( - d(y)x0)t+ \alpha (x0)\alpha (y)d(t) = - \alpha (y)d(t)x0 - d(y)tx0 for all y, t \in \scrN .
Putting u \circ v instead of t in the last expression, we get
d(y)( - x0)(u \circ v) = d(y)(u \circ v)( - x0) for all u, v, y \in \scrN . (3.12)
Replacing y by yt in (3.12) and using it again, we obtain
d(y)t( - x0)(u \circ v) + \alpha (y)d(t)( - x0)(u \circ v) = d(y)t(u \circ v)( - x0) + \alpha (y)d(t)(u \circ v)( - x0)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
42 A. BOUA, M. ASHRAF
for all y, t \in \scrN , which reduces to
d(y)\scrN
\bigl(
( - x0)(u \circ v) - (u \circ v)( - x0)
\bigr)
= \{ 0\} for all y, u, v \in \scrN .
Since d \not = 0, by 3-primeness of \scrN , we find that ( - x0)(u \circ v) = (u \circ v)( - x0) for all u, v \in \scrN ,
and in this case (3.10) implies ( - x)(u \circ v) = (u \circ v)( - x) for all u, v, x \in \scrN . Replacing x by - x
in the last equation, we obtain u \circ v \in Z(\scrN ) for all u, v \in \scrN . Now by Lemma 2.2, we conclude
that \scrN is a commutative ring. In this case, we obtain 2d(xy) = 0 for all x, y \in \scrN and by 2-torsion
freeness, we have d(xy) = 0 for all x, y \in \scrN . By definition of d, we get d(x)y + \alpha (x)d(y) = 0
for all x, y \in \scrN . Replacing y by yz in the above expression we obtain that d(x)yz = 0 for all
x, y, z \in \scrN , i.e., d(x)\scrN z = \{ 0\} for all x, z \in \scrN and by 3-primeness of \scrN we conclude that d = 0,
a contradiction.
Corollary 3.5. Let \scrN be a 2-torsion free 3-prime near-ring. Then there exists no nonzero deriva-
tion d such that d(x \circ y) = 0 for all x, y \in \scrN .
Corollary 3.6. Let \scrN be a 2-torsion free 3-prime near-ring. Then there exists no nonzero
semiderivation d associated with an onto map \alpha such that d(x \circ y) = 0 for all x, y \in \scrN .
Theorem 3.4. Let \scrN be a 2-torsion free 3-prime near-ring which admits a nonzero (1, \alpha )-
derivation d associated with an onto homomophism \alpha . Then the following assertions are equivalent:
(i) d([x, y]) = [x, y] for all x, y \in \scrN ,
(ii) \scrN is a commutative ring.
Proof. It is easy to verify that (ii) \Rightarrow (i).
(i) \Rightarrow (ii) Suppose that d([x, y]) = [x, y] for all x, y \in \scrN . Replacing y by xy, we get
[x, xy] = d([x, xy]) = d(x[x, y]) =
= d(x)[x, y] + \alpha (x)d([x, y]) for all x, y \in \scrN .
Since [x, xy] = x[x, y], the above expression becomes
d(x)[x, y] + \alpha (x)[x, y] = x[x, y] for all x, y \in \scrN .
Taking [u, v] instead of x and using our hypothesis, we arrive at
\alpha ([u, v])[[u, v], y] = 0 for all u, v, y \in \scrN .
This implies that
\alpha ([u, v])y[u, v] = \alpha ([u, v])[u, v]y for all u, v, y \in \scrN . (3.13)
Replacing y by yt in (3.13) and using it again, we get
\alpha ([u, v])yt[u, v] = \alpha ([u, v])[u, v]yt =
= \alpha ([u, v])y[u, v]t for all u, v, y, t \in \scrN ,
which forces that
\alpha ([u, v])\scrN [[u, v], t] = \{ 0\} for all u, v, t \in \scrN .
Since \scrN is 3-prime, we find that
\alpha ([u, v]) = 0 or [u, v] \in Z(\scrN ) for all u, v \in \scrN . (3.14)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
SOME ALGEBRAIC IDENTITIES IN 3-PRIME NEAR-RINGS 43
If there exist two elements u0, v0 \in \scrN such that [u0, v0] \in Z(\scrN ), then
d
\bigl(
[u0, v0][x, y]
\bigr)
= d
\bigl( \bigl[
[u0, v0]x, y
\bigr] \bigr)
=
= [u0, v0][x, y] for all x, y \in \scrN .
By definition of d, we find that
[u0, v0][x, y] = d([u0, v0][x, y]) =
= d([u0, v0])[x, y] + \alpha ([u0, v0])d([x, y]) =
= [u0, v0][x, y] + \alpha ([u0, v0])[x, y] for all x, y \in \scrN .
By the last expression, we obtain
\alpha ([u0, v0])xy = \alpha ([u0, v0])yx for all x, y \in \scrN . (3.15)
Replacing x by xt in (3.15) and using it again, we get
\alpha ([u0, v0])xty = \alpha ([u0, v0])yxt =
= \alpha ([u0, v0])xyt for all x, y, t \in \scrN .
Using this expression, we arrive at
\alpha ([u0, v0])\scrN [y, t] = \{ 0\} for all y, t \in \scrN .
Since \scrN is 3-prime, by Lemma 2.3, we obtain \alpha ([u0, v0]) = 0 or \scrN is a commutative ring. In this
case (3.14) becomes \alpha ([u, v]) = 0 for all u, v \in \scrN or \scrN is a commutative ring. Since \alpha is an onto
homomorphism, we find that \scrN is a commutative ring.
Theorem 3.5. Let \scrN be a 2-torsion free 3-prime near-ring. Then \scrN admits no nonzero (1, \alpha )-
derivation d associated with an onto homomorphism \alpha satisfying any one of the following conditions:
(i) d(x \circ y) = x \circ y for all x, y \in \scrN ,
(ii) d([x, y]) = x \circ y for all x, y \in \scrN ,
(iii) d(x \circ y) = [x, y] for all x, y \in \scrN .
Proof. (i) Suppose that d(x \circ y) = x \circ y for all x, y \in \scrN . Replacing x by xy, we have
x(x \circ y) = x \circ xy = d(x \circ xy) =
= d(x(x \circ y)) = d(x)(x \circ y) + \alpha (x)d(x \circ y) =
= d(x)(x \circ y) + \alpha (x)(x \circ y) for all x, y \in \scrN .
Putting u \circ v instead of x in the latter expression, we arrive at \alpha (u \circ v)
\bigl(
(u \circ v) \circ y
\bigr)
= 0 for all
u, v \in \scrN , which yields that
\alpha (u \circ v)y(u \circ v) = - \alpha (u \circ v)(u \circ v)y for all u, v, y \in \scrN . (3.16)
Replacing y by yt in (3.16) and using it again, we get
\alpha (u \circ v)yt(u \circ v) = - \alpha (u \circ v)(u \circ v)yt = \alpha (u \circ v)(u \circ v)y( - t) =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
44 A. BOUA, M. ASHRAF
= ( - \alpha (u \circ v)y(u \circ v))( - t) = \alpha (u \circ v)y( - (u \circ v))( - t) for all u, v, t \in \scrN .
This reduces to
\alpha (u \circ v)\scrN
\bigl(
- t( - u \circ v) + ( - u \circ v)t) = \{ 0\} for all u, v, t \in \scrN .
By 3-primeness of \scrN , we obtain
\alpha (u \circ v) = 0 or - u \circ v \in Z(\scrN ) for all u, v \in \scrN . (3.17)
Suppose there exist two elements u0, v0 \in \scrN such that - u0 \circ v0 \in Z(\scrN ), then
( - u0 \circ v0)(x \circ y) = (x( - u0 \circ v0) \circ y) = d((x( - u0 \circ v0) \circ y)) =
= d(( - u0 \circ v0)(x \circ y)) = d(( - u0 \circ v0))(x \circ y) + \alpha ( - u0 \circ v0)d(x \circ y) =
= ( - u0 \circ v0)(x \circ y) + \alpha ( - u0 \circ v0)(x \circ y) for all x, y \in \scrN .
This implies that
\alpha ( - u0 \circ v0)xy = - \alpha ( - u0 \circ v0)yx for all x, y \in \scrN . (3.18)
Replacing y by yt in (3.18) and using it again, we obtain
\alpha ( - u0 \circ v0)y(tx - ( - x)( - t)) = \{ 0\} for all x, y, t \in \scrN . (3.19)
Taking - x instead of x in (3.19), we get
\alpha ( - u0 \circ v0)\scrN ( - tx+ xt) = \{ 0\} for all x, t \in \scrN .
By 3-primeness of \scrN and Lemma 2.3, we deduce that \alpha ( - u0 \circ v0) = 0 or \scrN is a commutative ring.
Since \alpha is an additive map, (3.17) becomes
\alpha (u \circ v) = 0 for all u, v \in \scrN or \scrN is a commutative ring.
Using the fact that \alpha is onto homomorphism, we deduce that
u \circ v = 0 for all u, v \in \scrN or \scrN is a commutative ring.
By using Lemma 2.2, we conclude that \scrN is a commutative ring. Returning to our assumptions
and using 2-torsion freeness of \scrN , we obtain d(xy) = xy for all x, y \in \scrN . By definition of d, we get
d(x)y+\alpha (x)d(y) = xy for all x, y \in \scrN . Replacing x by xz, we obtain d(xz)y+\alpha (xz)d(y) = xzy
for all x, y, z \in \scrN , which means that \alpha (xz)d(y) = 0 for all x, y, z \in \scrN . Since \alpha is an onto
homomorphism, the last expression becomes xzd(y) = 0 for all x, y, z \in \scrN . Hence x\scrN d(y) = \{ 0\}
for all x, y \in \scrN . By 3-primeness of \scrN , we obtain that d = 0; a contradiction.
(ii) Assume that d([x, y]) = x \circ y for all x, y \in \scrN . Since \scrN is 2-torsion free, in particular, for
x = y, we find that x2 = 0 for all x \in \scrN . This implies that x(x+ y)2 = 0 for all x, y \in \scrN , hence
by a simple calculation, we obtain xyx = 0 for all x, y \in \scrN . By 3-primeness of \scrN , we conclude
that \scrN = \{ 0\} ; a contradiction.
(iii) Using the same techniques as used in (i) and (ii), we obtain the required result. The following
example shows that the existence of ”3-primeness” in the hypotheses of Theorems 3.2 and 3.3 is not
superfluous.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
SOME ALGEBRAIC IDENTITIES IN 3-PRIME NEAR-RINGS 45
Example. Let S be a zero symmetric left near-ring and
\scrN =
\left\{
\left( 0 a b
0 0 c
0 0 0
\right) a, b, c \in S
\right\} .
Then it can be easily seen that \scrN is a zero-symmetric left near-ring which is not 3-prime. Define
maps d, \alpha : \scrN \rightarrow \scrN such that
d
\left( 0 a b
0 0 0
0 c 0
\right) =
\left( 0 0 0
0 0 0
0 c 0
\right) , \alpha
\left( 0 a b
0 0 0
0 c 0
\right) =
\left( 0 a 0
0 0 0
0 0 0
\right) .
Then d is a (1, \alpha )-derivation satisfying d([x, y]) = 0 and d(x \circ y) = 0. However, \scrN is not a
commutative ring.
References
1. M. Ashraf, A. Shakir, On (\sigma , \tau )-derivations of prime near-rings-II, Sarajevo J. Math., 4, № 16, 23 – 30 (2008).
2. H. E. Bell, G. Mason, On derivations in near-rings. Near-rings and near-fields, North-Holland Math. Stud., 137
(1987).
3. H. E. Bell, On derivations in near-rings. II. Nearrings, nearfields and K-loops (Hamburg, 1995), Math. Appl., 426,
191 – 197 (1997).
4. H. E. Bell, A. Boua, L. Oukhtite, Semigroup ideals and commutativity in 3-prime near rings, Commun. Algebra. 43,
1757 – 1770 (2015).
5. A. Boua, L. Oukhtite, Semiderivations satisfying certain algebraic identities on prime near-rings, Asian-Eur. J. Math.,
6, № 3 (2013), 8 p.
6. G. Pilz, Near-rings, 2nd ed., 23, North Holland, Amsterdam (1983).
7. X. K. Wang, Derivations in prime near-rings, Proc. Amer. Math. Soc., 121, № 2, 361 – 366 (1994).
8. M. S.Samman, L. Oukhtite, A. Boua, A. Raji, Two sided \alpha -derivations in 3-prime near-rings, Rocky Mountain J.
Math., 46, № 4, 1379 – 1393 (2016).
Received 30.11.16,
after revision — 05.08.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
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| id | umjimathkievua-article-2324 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:13Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/c6/75ae9f13259544bbfe8e91242d5702c6.pdf |
| spelling | umjimathkievua-article-23242020-01-27T14:17:07Z Some algebraic identities in 3-prime near-rings Некоторые алгебраические тождества для 3-простіх почти колец Деякi алгебраїчнi тотожностi для 3 -простих майже кiлець Boua, A. Ashraf, M. Boua, A. Ashraf, M. Boua, A. Ashraf, M. прості майже кільця prime near-rings We extend the domain of applicability of the concept of $(1, \alpha)$-derivations in $3$-prime near-rings by analyzing the structure and commutativity of near-rings admitting $(1, \alpha)$-derivations satisfying certain differential identities. Розширено область застосовності поняття $(1, \alpha)$-похідних для $3$-простих майже кілець, як результат вивчення структури та комутативності майже кілець, що допускають $(1, \alpha)$-похідні, які задовольняють деякі диференціальні тотожності.   Розширено область застосовності поняття $(1, \alpha)$-похідних для $3$-простих майже кілець, як результат вивчення структури та комутативності майже кілець, що допускають $(1, \alpha)$-похідні, які задовольняють деякі диференціальні тотожності. Institute of Mathematics, NAS of Ukraine 2020-01-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2324 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 1 (2020); 36-45 Український математичний журнал; Том 72 № 1 (2020); 36-45 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2324/1547 Copyright (c) 2020 A. Boua ,M. Ashraf |
| spellingShingle | Boua, A. Ashraf, M. Boua, A. Ashraf, M. Boua, A. Ashraf, M. Some algebraic identities in 3-prime near-rings |
| title | Some algebraic identities in 3-prime near-rings |
| title_alt | Некоторые алгебраические тождества для 3-простіх почти колец Деякi алгебраїчнi тотожностi для 3 -простих майже кiлець |
| title_full | Some algebraic identities in 3-prime near-rings |
| title_fullStr | Some algebraic identities in 3-prime near-rings |
| title_full_unstemmed | Some algebraic identities in 3-prime near-rings |
| title_short | Some algebraic identities in 3-prime near-rings |
| title_sort | some algebraic identities in 3-prime near-rings |
| topic_facet | прості майже кільця prime near-rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2324 |
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