Some commutativity criteria for 3-prime near-rings
In the present paper, we introduce the notion of *-generalized derivation in near-ring N and investigate some properties involving that of *-generalized derivation of a *-prime near-ring N which forces N to be a commutative ring. Some properties of generalized semiderivations have also been given in...
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irk-123456789-1887542023-03-15T01:27:27Z Some commutativity criteria for 3-prime near-rings Raji, A. In the present paper, we introduce the notion of *-generalized derivation in near-ring N and investigate some properties involving that of *-generalized derivation of a *-prime near-ring N which forces N to be a commutative ring. Some properties of generalized semiderivations have also been given in the context of 3-prime near-rings. Consequently, some well known results have been generalized. Furthermore, we will give examples to demonstrate that the restrictions imposed on the hypothesis of various results are not superŕuous. 2021 Article Some commutativity criteria for 3-prime near-rings / A. Raji // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 280-298. — Бібліогр.: 10 назв. — англ. 1726-3255 DOI:10.12958/adm1439 2020 MSC: 16N60, 16W25, 16Y30 http://dspace.nbuv.gov.ua/handle/123456789/188754 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In the present paper, we introduce the notion of *-generalized derivation in near-ring N and investigate some properties involving that of *-generalized derivation of a *-prime near-ring N which forces N to be a commutative ring. Some properties of generalized semiderivations have also been given in the context of 3-prime near-rings. Consequently, some well known results have been generalized. Furthermore, we will give examples to demonstrate that the restrictions imposed on the hypothesis of various results are not superŕuous. |
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Raji, A. |
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Raji, A. Some commutativity criteria for 3-prime near-rings Algebra and Discrete Mathematics |
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Raji, A. |
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Raji, A. |
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Some commutativity criteria for 3-prime near-rings |
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Some commutativity criteria for 3-prime near-rings |
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Some commutativity criteria for 3-prime near-rings |
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Some commutativity criteria for 3-prime near-rings |
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Some commutativity criteria for 3-prime near-rings |
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some commutativity criteria for 3-prime near-rings |
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Інститут прикладної математики і механіки НАН України |
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2021 |
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http://dspace.nbuv.gov.ua/handle/123456789/188754 |
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Some commutativity criteria for 3-prime near-rings / A. Raji // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 280-298. — Бібліогр.: 10 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT rajia somecommutativitycriteriafor3primenearrings |
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2025-07-16T10:57:32Z |
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2025-07-16T10:57:32Z |
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1837800832984154112 |
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© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 32 (2021). Number 2, pp. 280ś298
DOI:10.12958/adm1439
Some commutativity criteria
for 3-prime near-rings
A. Raji
Communicated by G. Pilz
Abstract. In the present paper, we introduce the notion of ∗-
generalized derivation in near-ring N and investigate some properties
involving that of ∗-generalized derivation of a ∗-prime near-ring
N which forces N to be a commutative ring. Some properties of
generalized semiderivations have also been given in the context of 3-
prime near-rings. Consequently, some well known results have been
generalized. Furthermore, we will give examples to demonstrate
that the restrictions imposed on the hypothesis of various results
are not superŕuous.
1. Introduction
Throughout this paper N will be a near-ring with multiplicative center
Z(N). A near-ring N is said to be 3-prime if xNy = {0} for x, y ∈ N
implies x = 0 or y = 0. It is called 3-semiprime if xNx = {0} implies
x = 0. Recalling that N is called 2-torsion free if (N,+) has no elements
of order 2; and N is called zero-symmetric if 0 ·x = x · 0 = 0 for all x ∈ N .
For any x, y ∈ N ; as usual [x, y] = xy − yx and x ◦ y = xy + yx will
denote the well-known Lie and Jordan products respectively. An additive
mapping ∗ : N → N is called an involution if (xy)∗ = y∗x∗ and (x∗)∗ = x
for all x, y ∈ N . A near-ring N equipped with an involution ∗ is said to
be ∗-prime if xNy = xNy∗ = {0} implies x = 0 or y = 0. Obviously a left
2020 MSC: 16N60, 16W25, 16Y30.
Key words and phrases: 3-prime near-rings, 3-semiprime near-rings, involution,
∗-derivation, semiderivation, commutativity.
https://doi.org/10.12958/adm1439
A. Raji 281
(resp. right) near-ring with involution is in fact distributive, and hence
also zero-symmetric. Indeed, let N be a left near-ring with involution ∗,
and let x, y, z,∈ N . Then
((x+ y)z)∗ = z∗(x∗ + y∗)
= z∗x∗ + z∗y∗.
Applying ∗ to both sides, we obtain
(x+ y)z = (z∗x∗ + z∗y∗)∗
= x∗∗z∗∗ + y∗∗z∗∗
= xz + yz.
Clearly, a near-ring with an involution is not necessarily a ring. Indeed, in
Example 1, since the addition law in N is not commutative, N cannot be
a ring.
The commutativity of 3-prime near-rings with derivation was initiated by
Bell and Mason in [3]. Over the last two decades, a lot of work has been done
on this subject. Recently, in [2], Ashraf and Siddeeque deőned the following
notations: An additive mapping d : N → N is called a ∗-derivation if there
exists an involution ∗ : N → N such that d(xy) = d(x)y∗ + xd(y), for all
x, y ∈ N . An additive mapping F : N → N is called a left ∗-multiplier if
F (xy) = F (x)y∗ hold for all x, y ∈ N . Motivated by these concepts, we
introduce the concepts of ∗-generalized derivation in near-rings as follows:
An additive mapping F : N → N is called a ∗-generalized derivation if
there exist a ∗-derivation d of N such that
F (xy) = F (x)y∗ + xd(y) for all x, y ∈ N.
One may observe that the concept of ∗-generalized derivation includes
the concept of ∗-derivation and left ∗-multiplier. Hence it should be
interesting to extend some results concerning these notions to ∗-generalized
derivation. An additive mapping δ : N → N is said to be a derivation if
δ(xy) = xδ(y)+δ(x)y for all x, y ∈ N , or equivalently, as noted in [10], that
δ(xy) = δ(x)y + xδ(y) for all x, y ∈ N . An additive mapping f : N → N
is said to be a generalized derivation on N if there exists a derivation
δ : N → N such that f(xy) = f(x)y + xδ(y) for all x, y ∈ N . An
additive mapping d : N → N is called semiderivation if there is a function
g : N → N such that d(xy) = xd(y) + d(x)g(y) = g(x)d(y) + d(x)y
and d(g(x)) = g(d(x)) for all x, y ∈ N , or equivalently, as noted in [5],
that d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and d(g(x)) = g(d(x))
282 Commutativity criteria for 3-prime near-rings
for all x, y ∈ N . Obviously, any derivation is a semiderivation, but the
converse is not true in general (see [5]). An additive mapping F : N → N
is called generalized semiderivation if there is a semiderivation d : N → N
associated with a function g : N → N such that
F (xy) = F (x)y + g(x)d(y) = d(x)g(y) + xF (y) for all x, y ∈ N.
Of course any semiderivation is a generalized semiderivation. Moreover, if
g is the identity map of N , then all generalized semiderivations associated
with g are merely generalized derivations of N . Recently, many researchers
have studied commutativity of prime and semiprime rings admitting
suitably constrained additive mappings, as: automorphisms, derivations
and involutions acting on appropriate subsets of the rings (see, for example,
Bell and Mason [3]; Bresar and Vukman [6]; Fong [7]; Oukhtite and
Salhi [8]). Being motivated by their invaluable research, it is natural then
to ask if we can apply the involution so as to study the structure of
a near-ring. In this paper, őrstly we would like to study the structure
of a ∗-prime left near-ring and 3-semiprime left near-rings as having an
involution. More precisely, we shall prove that a ∗-prime near-ring must
be a commutative ring. Secondly, some results concerning composition of
generalized semiderivation on a 3-prime right near-ring will be exposed.
As for terminologies used here without mention, we refer to Pilz [9].
2. Main results
In [6] M. Bresar and J.Vukman introduced the notion of ∗-derivation
(resp. reverse ∗-derivation) in rings and proved that if a prime ∗-ring
admits a nonzero ∗-derivation (resp. reverse ∗-derivation), then the ring is
commutative. This result was extended to 3-semiprime ring by S. Ali [1],
who proved that if a 3-semiprime ∗-ring R admits a ∗-derivation (resp.
reverse ∗-derivation) d, then d maps R into Z(R) where Z(R) is the center
of R. Further, Ashraf and Seddeeque [2] showed that a 3-prime ∗-near-ring
N with a nonzero ∗-derivation d, must be commutative. Motivated by
these results mentioned above, we succeeded in establishing the following
results:
Theorem 1. Let N be a ∗-prime near ring. If N admits a nonzero
∗-generalized derivation F , then N is a commutative ring.
A. Raji 283
Proof. For all x, y, z ∈ N , we have
F (xyz) = F (xy)z∗ + xyd(z)
= (F (x)y∗ + xd(y))z∗ + xyd(z)
= F (x)y∗z∗ + xd(y)z∗ + xyd(z).
On the other hand,
F (xyz) = F (x)(yz)∗ + xd(yz)
= F (x)z∗y∗ + x(d(y)z∗ + yd(z))
= F (x)z∗y∗ + xd(y)z∗ + xyd(z).
Combining the both expressions for F (xyz), we get
F (x)y∗z∗ = F (x)z∗y∗ for allx, y, z ∈ N. (1)
Replacing y and z by y∗ and z∗, respectively, in (1), we obtain
F (x)yz = F (x)zy for allx, y, z ∈ N. (2)
Taking yt instead of y in (2), where t ∈ N, and using it again we arrive at
F (x)yzt = F (x)ytz for allx, y, z, t ∈ N. (3)
So that,
F (x)y[z, t] = 0 for allx, y, z, t ∈ N. (4)
Replacing z and t by t∗ and z∗, respectively, in (4), we obtain
F (x)y[t∗, z∗] = 0 for allx, y, z, t ∈ N. (5)
Since [t∗, z∗] = [z, t]∗, equation (5) can be written as
F (x)y[z, t]∗ = 0 for allx, y, z, t ∈ N. (6)
From (4) and (6), we őnd that
F (x)N [z, t] = F (x)N [z, t]∗ = {0} for all x, z, t ∈ N. (7)
In view of the ∗-primeness of N , (7) shows that
F (x) = 0 or [z, t] = 0 for all x, z, t ∈ N.
Since F ̸= 0, we conclude that [z, t] = 0 for z, t ∈ N . And therefore, the
multiplicative law of N is commutative. Now, let x, y, z ∈ N . Then we
284 Commutativity criteria for 3-prime near-rings
have (x+ x)(y+ z) = (y+ z)(x+x). It follows that (x+ x)y+ (x+x)z =
(y+z)x+(y+z)x for all x, y, z ∈ N. The previous relation can be rewritten
as y(x+ x) + z(x+ x) = x(y + z) + x(y + z). After simplifying, we have
yx+ zx = xz + xy which implies that
x((y + z)− (z + y)) = 0 for all x, y, z ∈ N. (8)
Replacing y and z by y∗ and z∗, respectively, in (8), we arrive at
x((y + z)− (z + y))∗ = 0 for all x, y, z ∈ N. (9)
Now, comparing the identities (8) and (9), we őnd that
x((y + z)− (z + y)) = x((y + z)− (z + y))∗ = 0 for all x, y, z ∈ N.
Next putting xt, where t ∈ N , in place of x, we obtain
xt((y + z)− (z + y)) = xt((y + z)− (z + y))∗ = 0 for all x, y, z, t ∈ N.
It follows that
xN((y + z)− (z + y)) = xN((y + z)− (z + y))∗ = {0} for all x, y, z ∈ N.
(10)
In the light of the ∗-primeness of N and N ̸= {0}, (10) shows that
(y + z)− (z + y) = 0 for all y, z ∈ N , i.e, (N,+) is abelian. Consequently,
N is a commutative ring. This completes the proof of our theorem.
The following example proves that the hypothesis of ∗-primeness in
the previous Theorem is not superŕuous.
Example 1. Let N =
{(
0 0 x
0 0 y
0 0 0
)
| 0, x, y ∈ S
}
, where S is a zero-
symmetric left near-ring such that the addition in S is not commutative.
Of course N with matrix addition and matrix multiplication is a left
near-ring. Deőne mappings ∗, d, F : N → N such that
(
0 0 x
0 0 y
0 0 0
)
∗
=
(
0 0 y
0 0 x
0 0 0
)
, d
(
0 0 x
0 0 y
0 0 0
)
=
(
0 0 x
0 0 0
0 0 0
)
and F
(
0 0 x
0 0 y
0 0 0
)
=
(
0 0 0
0 0 y
0 0 0
)
.
It is clear that N is not ∗-prime near-ring and F is a nonzero ∗-generalized
derivation associated with a ∗-derivation d. But, since the addition in N
is not commutative, N cannot be a commutative ring.
A. Raji 285
Theorem 2. Let N be a 3-semiprime near-ring. If N admits a nonzero
∗-generalized derivation F , then N contains a nonzero commutative ring
under the operations of N.
Proof. Given that there exists a ∗-generalized derivation F of N . Arguing
as in the proof of Theorem 1, we have relation (4)
F (x)y[z, t] = 0 for all x, y, z, t ∈ N. (11)
Taking z = F (x) and y = ty in (11) and applying ∗, we őnd that
[F (x), t]∗y∗t∗(F (x))∗ = 0 for all x, y, t ∈ N. (12)
On the other hand, left multiplying (11) by t and replacing z by F (x), we
obtain
tF (x)y[F (x), t] = 0 for all x, y, t ∈ N.
Now, applying ∗ for this relation, we get
[F (x), t]∗y∗(F (x))∗t∗ = 0 for all x, y, t ∈ N. (13)
Combining the identities (12) and (13), we conclude that
[F (x), t]∗y∗[t∗, (F (x))∗] = 0 for all x, y, t ∈ N,
which means that
[F (x), t]∗y∗[F (x), t]∗ = 0 for all x, y, t ∈ N. (14)
Once again applying ∗ to (14), we obtain
[F (x), t]y[F (x), t] = 0 for all x, y, t ∈ N.
Accordingly,
[F (x), t]N [F (x), t] = {0} for all x, t ∈ N.
In view of the semiprimeness of N , the last relation assures that [F (x), t]
= 0 for all x, t ∈ N ; in other words F (N) ⊆ Z(N). As F ̸= 0, then there
is an element x0 ∈ N such that F (x0) ̸= 0. Now let H the subset of N be
deőned by:
H = {F (x0)r | r ∈ N}.
Our goal is to show that H is a commutative ring included in N . Firstly,
we have H ̸= {0}. Indeed, suppose that F (x0)r = 0 for all r ∈ N . Right
286 Commutativity criteria for 3-prime near-rings
multiplying by F (x0) we get F (x0)rF (x0) = 0 for all r ∈ N . Using the
semiprimeness of N , we őnd that F (x0) = 0; a contradiction. Secondly,
showing that (H,+, .) is a semiprime left near-ring. As N is a left near-ring
and F (x0) ∈ Z(N), then with a simple calculation, it is easy to see that H
is a nonzero left near-ring. Next, we show that H is a semiprime near-ring.
Indeed, let a ∈ N such that
F (x0)aHF (x0)a = {0}.
Then
F (x0)aF (x0)rF (x0)a = 0 for all r ∈ N. (15)
Right multiplying (15) by F (x0) and using the fact that F (x0) ∈ Z(N),
we obtain
F (x0)aF (x0)rF (x0)aF (x0) =
(
(
F (x0)
)2
a
)
r
(
(
F (x0)
)2
a
)
= 0
for all r ∈ N . It follows that
(
(
F (x0)
)2
a
)
N
(
(
F (x0)
)2
a
)
= {0}.
In view of the semiprimeness of N , the preceding equation shows that
(
F (x0)
)2
a = 0 . Once again left multiplying this equation by ar, where
r ∈ N , we get
F (x0)arF (x0)a = 0 for all r ∈ N. (16)
Since N is a semiprime near-ring, (16) implies that F (x0)a = 0. And
therefore, H is a 3-semiprime left near-ring. Now, returning to equation
(2) of the previous theorem, this equation remains valid in this theorem.
So we have F (x)[y, z] = 0 for all x, y, z ∈ N . Taking x = x0 and left
multiplying by r, we obtain F (x0)r[y, z] = 0 for all r, y, z ∈ N . Setting
t = F (x0)r, where r ∈ N , then the last relation yields t[y, z] = 0 for all
t ∈ H, y, z ∈ N . In particular, for y, z ∈ H, we have t[y, z] = 0 for all
t, y, z ∈ H which implies that
[y, z]t[y, z] = 0 for all t, y, z ∈ H.
Accordingly,
[y, z]H[y, z] = {0} for all y, z ∈ H.
Using the fact that H is a 3-semiprime near-ring, the last equation shows
that [y, z] = 0 for all y, z ∈ H. Hence, the multiplicative law of H is
commutative; thereby, for all x, y, z ∈ H we have (x+x)(y+z) = (y+z)(x+
A. Raji 287
x). Arguing as in the proof of Theorem 1, we arrive at x((y+z)−(z+y)) = 0
for all x, y, z ∈ H. Replacing x by ((y + z)− (z + y))x, we get
((y + z)− (z + y))x((y + z)− (z + y)) = 0 for all x, y, z ∈ H.
Consequently, ((y+z)− (z+y))H((y+z)− (z+y)) = {0} for all y, z ∈ H
and semiprimeness of H yields that (y + z)− (z + y) = 0 for all y, z ∈ H .
So, we conclude that H is a commutative ring and hence N contains
a nonzero commutative ring. Then, the Theorem is proved.
Theorem 3. Let N be a 3-semiprime near-ring and F be a ∗-generalized
derivation of N associated with a ∗-derivation d which commutes with ∗.
Then F must be a left ∗-multiplier if F 2 = 0.
Proof. By the hypothesis, we have F 2(xy) = F (x)d(y∗) + F (x)(d(y))∗ +
xd2(y) = 0 for all x, y ∈ N . Replacing x by F (x) in the preceding equation,
we obtain
F (x)d2(y) = 0 for all x, y ∈ N. (17)
By deőning F , we have F (xyz∗) = F (xy)z + xyd(z∗) and F (xyz∗) =
F (x)zy∗ + xd(y)z + xyd(z∗) for all x, y, z ∈ N . Comparing the above
expressions of F (xyz∗), we őnd that F (xy)z = F (x)zy∗ + xd(y)z for all
x, y, z ∈ N . Now, taking xt instead of x in (17) and applying the last
result, we arrive at
xd(t)d2(y) = 0 for all x, t, y ∈ N.
Writing d(y) in the place of t and using ([2], Theorem 3.9), we get
d2(y)xd2(y) = 0 for all x, y ∈ N . By the semiprimeness of N , we con-
clude that d2 = 0, furthermore d = 0 by ([3], Lemma 2.5). And there-
fore, F (xy) = F (x)y∗ for all x, y ∈ N which means that F is a left
∗-multiplier.
3. Conditions involving generalized semiderivations
In this section, our near-rings are right near-rings. In order to prove
our main theorems, we shall need the following lemmas.
Lemma 1. Let N be a 3-prime near-ring.
(i) ([4], Lemma 1.2 (iii)) If z ∈ Z(N)\{0} and xz ∈ Z(N), then x ∈ Z(N).
(ii) ([4], Lemma 1.5) If N ⊆ Z(N), then N is a commutative ring.
288 Commutativity criteria for 3-prime near-rings
Lemma 2. Let N be an arbitrary right near-ring admitting a semideriva-
tion d associated with a function g. Then N is a zero-symmetric near-ring.
Moreover, if N is 3-prime and d ̸= 0, then g(0) = 0.
Proof. Since N is a right near-ring, we have 0 · x = 0 for all x ∈ N . Now,
let d be a semiderivation of N associated with a function g and let x, y
be two arbitrary elements of N , by deőning the property of d, we have
d(x · 0) = xd(0) + d(x)g(0) = x · 0 + d(x)g(0)
= g(x)d(0) + d(x) · 0 = g(x) · 0 + d(x) · 0
and
d((x · 0) · y) = x · 0 · d(y) + d(x · 0)g(y) = x · 0 + d(x · 0)g(y)
= x · 0 + (x · 0 + d(x)g(0))g(y)
= x · 0 + (g(x) · 0 + d(x) · 0)g(y).
So, d((x · 0) · y) = x · 0+ x · 0 · g(y) + d(x)g(0)g(y) = x · 0+ g(x) · 0 · g(y) +
d(x) · 0 · g(y) which implies that x · 0 + d(x)g(0)g(y) = g(x) · 0 + d(x) · 0.
As g(x) · 0 + d(x) · 0 = x · 0 + d(x)g(0) by deőning d(x · 0), and after
simplifying, the preceding result shows that
d(x)g(0)g(y) = d(x)g(0) for all x, y ∈ N.
On the other hand, we have
d(x · (0 · y)) = xd(0 · y) + d(x)g(0 · y) = x · 0 + d(x)g(0) for all x, y ∈ N.
From the preceding expressions of d(x ·0 ·y), we őnd that x ·0+d(x)g(0) =
x · 0 + x · 0 + d(x)g(0)g(y) for all x, y ∈ N. After simplifying, the latter
equation assures that x · 0 = 0 for all x ∈ N. Consequently, N is a zero-
symmetric near-ring. Assume now that N is 3-prime and d ≠ 0, by applying
the last result, we get d(x·0) = 0 = x·0+d(x)g(0) = d(x)g(0) for all x ∈ N.
Replacing x by xt gives d(xt)g(0) = 0 =
(
g(x)d(t) + d(x)t
)
g(0) for all
x, t ∈ N , which implies that d(x)tg(0) = 0 for all x, t ∈ N. Accordingly,
d(x)Ng(0) = {0} for all x ∈ N . Since N is 3-prime and d ̸= 0, we conclude
that g(0) = 0.
In right near-ring N , left distributive property does not hold in general.
However, the following Lemma has its own signiőcance.
A. Raji 289
Lemma 3. Let N be a near-ring admitting a semiderivation d associated
with a function g such that g(xy) = g(x)g(y) for all x, y ∈ N . Then N
satisőes the following partial distributive law:
x(yd(z) + d(y)g(z)) = xyd(z) + xd(y)g(z) for all x, y, z ∈ N.
Proof. We have d((xy)z) = xyd(z) + d(xy)g(z) = xyd(z) + xd(y)g(z) +
d(x)g(y)g(z) and d(x(yz)) = xd(yz) + d(x)g(yz) = x(yd(z) + d(y)g(z)) +
d(x)g(yz). Comparing these two equations gives the desired result.
Lemma 4. Let N be a near-ring. If N admits a semiderivation d associ-
ated with an onto map g, then d(z) ∈ Z(N) for all z ∈ Z(N).
Proof. Calculate d(xz) and d(zx) where x ∈ N, z ∈ Z(N) and compare.
Theorem 4. Let N be a 2-torsion free 3-prime near-ring admitting
a generalized semiderivation F associated with a nonzero semiderivation d
which is associated with an onto map g such that g(xy) = g(x)g(y) for all
x, y ∈ N . If F ([x, y]) = 0 for all x, y ∈ N , then N is a commutative ring.
Proof. By the hypothesis given, we have F (xy) = F (yx) for all x, y ∈ N ,
hence
F (x)y + g(x)d(y) = d(y)g(x) + yF (x) for all x, y ∈ N.
Replacing x by [r, s] in the previous equation, we get
g([r, s])d(y) = d(y)g([r, s]) for all y, r, s ∈ N. (18)
Substituting yd(t) for y in (18) and invoking Lemma 3, we obtain
g([r, s])yd2(t) + g([r, s])d(y)g(d(t)) = yd2(t)g([r, s]) + d(y)g(d(t))g([r, s])
for all y, r, s, t ∈ N. Taking into account that g(d(t)) = d(g(t)) for all
t ∈ N and using (18), we őnd that
g([r, s])yd2(t) = yd2(t)g([r, s]) for all y, r, s, t ∈ N. (19)
Replacing y by yn in (19) and using (19) again, we get
g([r, s])ynd2(t) = ynd2(t)g([r, s])
= y
(
nd2(t)g([r, s])
)
= yg([r, s])nd2(t) for all y, n, r, s, t ∈ N,
290 Commutativity criteria for 3-prime near-rings
which leads to
[g([r, s]), y]Nd2(t) = {0} for all y, r, s, t ∈ N. (20)
By the 3-primeness of N , equation (20) assures that
d2(t) = 0 or g([r, s]) ∈ Z(N) for all r, s, t ∈ N. (21)
Suppose that d2(t) = 0 for all t ∈ N . In particular, we have
0 = d2(tu)
= d(td(u) + d(t)g(u))
= 2d(t)d(g(u)) for all t, u ∈ N.
In view of the 2-torsion freeness of N , the above relation implies that
d(t)d(g(u)) = 0 for all t, u ∈ N . Now, replacing t by xt, where x ∈ N , in
the preceding equation, we őnd that d(x)td(g(u)) = 0 for all x, t, u ∈ N . It
follows that d(x)Nd(g(u)) = {0} for all x, u ∈ N which, as N is 3-prime
and d ≠ 0, implies that d(g(u)) = 0 = g(d(u)) for all u ∈ N . Now, by the
deőning property of d, we have
d(vd(u)) = vd2(u) + d(v)g(d(u))
= g(v)d2(u) + d(v)d(u) for all u, v ∈ N.
Comparing these two expressions for d(vd(u)) gives d(v)d(u) = 0 for all
u, v ∈ N. Taking v = vt and using the same arguments as used above,
we conclude that d = 0. But this contradicts our assumption that d ̸= 0;
hence, equation (21) forces g([r, s]) ∈ Z(N) for all r, s ∈ N . Replacing s
by sr in the previous result and noting that [r, sr] = [r, s]r, we obtain
g([r, sr]) = g([r, s]r) = g([r, s])g(r) ∈ Z(N) for all r, s ∈ N.
According to Lemma 1(i), we conclude that
g([r, s]) = 0 or g(r) ∈ Z(N) for all r, s ∈ N. (22)
Suppose there exist two elements r, s ∈ N such that g([r, s]) = 0. In this
case, using Lemma 2 and g is onto, we őnd that [r, s] = 0 which means
that rs = sr. And therefore g(rs) = g(sr), hence g(r)g(s) = g(s)g(r). So,
equation (22) reduces to
g(r)g(s) = g(s)g(r) or g(r) ∈ Z(N) for all r, s ∈ N. (23)
A. Raji 291
But it is clear that g(r) ∈ Z(N) implies g(r)g(s) = g(s)g(r) for all s ∈ N .
Consequently, (23) yields g(r)g(s) = g(s)g(r) for all r, s ∈ N . Since g is
onto, the latter equation shows that rs = sr for all r, s ∈ N. Which means
that N ⊆ Z(N) and our result follows by Lemma 1(ii).
The following result proves that the conclusion of the previous Theorem
is not valid if we replace the product [x, y] by x ◦ y. Indeed,
Theorem 5. Let N be a 2-torsion free 3-prime near-ring. There is no
nonzero generalized semiderivation F associated with a semiderivation d
which is associated with an onto map g such that g(xy) = g(x)g(y) for all
x, y ∈ N satisőes F (x ◦ y) = 0 for all x, y ∈ N.
Proof. Suppose that
F (x ◦ y) = 0 for all x, y ∈ N. (24)
It follows that
F (xy) + F (yx) = 0 for all x, y ∈ N.
By the deőning property of F , we get
F (x)y + g(x)d(y) + d(y)g(x) + yF (x) = 0 for all x, y ∈ N. (25)
Substituting u ◦ v for x in (25) and invoking (24), we obtain
g(u ◦ v)d(y) + d(y)g(u ◦ v) = 0 for all y, u, v ∈ N. (26)
Taking yt instead of y in (26) and using Lemma 3, we get
g(u ◦ v)yd(t) + g(u ◦ v)d(y)g(t) + d(y)g(t)g(u ◦ v) + yd(t)g(u ◦ v) = 0
for all u, v, y, t ∈ N. Replacing t by u ◦ v and using (26), we arrive at
g(u ◦ v)yd(u ◦ v) = −yd(u ◦ v)g(u ◦ v) for all u, v, y ∈ N. (27)
Substituting yt for y in (27), we őnd that
g(u ◦ v)ytd(u ◦ v) = −ytd(u ◦ v)g(u ◦ v)
= (−y)(td(u ◦ v)g(u ◦ v))
= (−y)(−g(u ◦ v)td(u ◦ v))
= (−y)(−g(u ◦ v))td(u ◦ v) for all u, v, y, t ∈ N.
292 Commutativity criteria for 3-prime near-rings
Hence,
(
g(u ◦ v)y − (−y)(−g(u ◦ v))
)
td(u ◦ v) = 0 for all u, v, y, t ∈ N. (28)
Therefore, (28) can be rewritten as
(
− (−g(u ◦ v))y+ y(−g(u ◦ v))
)
Nd(u ◦ v) = {0} for all u, v, y ∈ N. (29)
By 3-primeness of N , equation (29) implies that
−g(u ◦ v) ∈ Z(N) or d(u ◦ v) = 0 for all u, v ∈ N. (30)
So,
−g(u ◦ v) ∈ Z(N) or g(d(u ◦ v)) = 0 for all u, v ∈ N. (31)
Suppose there are two elements u0 and v0 of N such that −g(u0 ◦ v0) ∈
Z(N). If d(Z(N)) = {0}, then 0 = d(−g(u0 ◦ v0)) = −d(g(u0 ◦ v0)) =
d(g(u0 ◦ v0)) which implies that g(d(u0 ◦ v0)) = 0. On the other hand,
if d(Z(N)) ̸= {0}, in this case, returning to (26) and replacing y by an
element z0 ∈ Z(N) such that d(z0) ̸= 0, also taking u = u0 and v = v0,
according to Lemma 4, we obtain 2g(u0 ◦ v0)d(z0) = 0. By 2-torsion
freeness the latter relation shows that
g(u0 ◦ v0)d(z0) = 0 (32)
Right multiplying (32) by n, where n ∈ N , we obtain
g(u0 ◦ v0)nd(z0) = 0 for all n ∈ N.
Which implies that
g(u0 ◦ v0)Nd(z0) = {0}.
In view of the 3-primeness of N and d(z0) ̸= 0, we conclude that g(u0◦v0) =
0, hence d(g(u0 ◦v0)) = 0 which yields g(d(u0 ◦v0)) = 0. Thus, in the both
cases, i.e d(Z(N)) = {0} or d(Z(N)) ̸= {0}, we őnd that g(d(u0 ◦v0)) = 0.
Consequently, (31) reduces to
g(d(u ◦ v)) = 0 for all u, v ∈ N.
According to Lemma 2 and g is onto, the last relation yields
d(u ◦ v) = 0 for all u, v ∈ N. (33)
A. Raji 293
Taking vu instead of v in (33) and using (33), we get
(u ◦ v)d(u) = 0 for all u, v ∈ N
that is,
uvd(u) = −vud(u) for all u, v ∈ N.
Next putting vt, where t ∈ N , in place of v we arrive at
uvtd(u) = (−v)(−u)td(u) for all u, v, t ∈ N.
It follows that
(uv + v(−u))td(u) = 0 for all u, v, t ∈ N.
Taking −u instead of u, hence the last relation can be rewritten as
(−uv + vu)Nd(−u) = {0} for all u, v ∈ N.
Which, in the light of the primeness of N , yields
u ∈ Z(N) or d(u) = 0 for all u ∈ N. (34)
Let u0 ∈ Z(N), from (33) and 2-torsion freeness, we have d(u0 ◦ v) = 0 =
d(u0v) for all v ∈ N . By deőning d, we get g(u0)d(v) + d(u0)v = 0 for all
v ∈ N . Replacing v by vu0 in the last result, we őnd that d(u0)vu0 = 0
for all v ∈ N . Once again using the 3-primeness of N , we conclude that
d(u0) = 0 or u0 = 0, given that d(u0) = 0. And therefore, (34) proves that
d(u) = 0 for all u ∈ N .
Our goal in what follows is to show that d = 0 implies F = 0. Let
x, y ∈ N , by deőning F and our hypothesis d = 0, equation (24) yields
F (x◦y) = 0 = F (xy)+F (yx) = xF (y)+yF (x) for all x, y ∈ N . Replacing
x by x ◦ t, we get (x ◦ t)F (y) = 0 for all x, y, t ∈ N , which means that
xtF (y) = −txF (y) for all x, y, t ∈ N.
Taking tm instead of t in the last equation, where m ∈ N , and using the
same equation we arrive at
xtmF (y) = (−t)(−x)mF (y) for all x, y, t,m ∈ N.
It follows that
(xt+ t(−x))NF (y) = {0} for all x, y, t ∈ N.
294 Commutativity criteria for 3-prime near-rings
Putting −x instead of x, we get
(−xt+ tx)NF (y) = {0} for all x, y, t ∈ N.
In the light of the 3-primeness of N , we conclude that N ⊆ Z(N) or F = 0.
But if N ⊆ Z(N), N is a commutative ring by Lemma 1(ii). In this case,
returning to (24) and using the fact that N is 2-torsion free, we obtain
F (xy) = 0 = xF (y) for all x, y ∈ N . Replacing x by xt, we get xtF (y) = 0
for all x, y, t ∈ N . Once again N is 3-prime, the last expression yields
F = 0, which is contrary to our hypothesis. This completes the proof of
our Theorem.
Theorem 6. Let N be a 2-torsion free 3-prime near-ring admitting
a generalized semiderivation F associated with a nonzero semiderivation d
which is associated with an automorphism map g such that F ([x, y]) = [x, y]
for all x, y ∈ N , then N is a commutative ring.
Proof. Assume that
F ([x, y]) = [x, y] for all x, y ∈ N. (35)
Replacing y by yx in (35), we get
F ([x, y])x+ g([x, y])d(x) = [x, y]x for all x, y ∈ N. (36)
In view of (35) and g is an homomorphism, equation (36) can be rewritten
as
g(x)g(y)d(x) = g(y)g(x)d(x) for all x, y ∈ N.
Once again, since g is onto, the last equation yields
g(x)yd(x) = yg(x)d(x) for all x, y ∈ N. (37)
Taking yt instead of y in (37) and using (37) again, we őnd that
g(x)ytd(x) = ytg(x)d(x)
= y(tg(x)d(x))
= yg(x)td(x) for all x, y, t ∈ N
implying that
(g(x)y − yg(x))td(x) = 0 for all x, y, t ∈ N
A. Raji 295
that is,
[g(x), y]Nd(x) = {0} for all x, y ∈ N. (38)
By the use of 3-primeness of N , (38) yields
g(x) ∈ Z(N) or d(x) = 0 for all x ∈ N (39)
in the latter case, we see that if d(x) = 0, then g(d(x)) = d(g(x)) = 0. So,
(39) proves that either
g(x) ∈ Z(N) or d(g(x)) = 0 for all x ∈ N.
Since g is onto, the above relation shows that x ∈ Z(N) or d(x) = 0 for
all x ∈ N . And according to Lemma 4 we conclude that d(x) ∈ Z(N) for
all x ∈ N . Replacing x by xy, gives
d(xy) = xd(y) + d(x)g(y) ∈ Z(N) for all x, y ∈ N. (40)
Using Lemma 3, (40) implies that
x2d(x) + xd(x)g(y) = x2d(x) + d(x)g(y)x for all x, y ∈ N,
this is reduced to
[g(y), x]Nd(x) = {0} for all x, y ∈ N. (41)
Since N is 3-prime and g is onto, (41) shows that
x ∈ Z(N) or d(x) = 0 for all x ∈ N. (42)
If there is an element x0 of N such that d(x0) = 0, according to the
equation (40), we obtain
x0d(y) ∈ Z(N) for all y ∈ N. (43)
Since d ̸= 0, then Lemma 1(i) assures that x0 ∈ Z(N), and therefore (42)
reduces to x ∈ Z(N) for all x ∈ N . The Lemma 1(ii) demonstrates that
N is a commutative ring.
Theorem 7. Let N be a 2-torsion free 3-prime near-ring. There is no
generalized semiderivation F associated with a nonzero semiderivation d
which is associated with an automorphism map g such that F (x◦y) = x◦y
for all x, y ∈ N .
296 Commutativity criteria for 3-prime near-rings
Proof. Suppose that there is F which indicates the following
F (x ◦ y) = x ◦ y for all x, y ∈ N. (44)
Substituting yx for y in (44), because of x ◦ yx = (x ◦ y)x, we obtain
F ((x ◦ y)x) = (x ◦ y)x for all x, y ∈ N.
By deőning F , we get
F (x ◦ y)x+ g(x ◦ y)d(x) = (x ◦ y)x for all x, y ∈ N. (45)
From (44) and (45), we őnd that
g(x ◦ y)d(x) = 0 for all x, y ∈ N.
As g is an homomorphism, we have
g(x)g(y)d(x) = −g(y)g(x)d(x) for all x, y ∈ N.
Since g is onto, the last equation shows that
g(x)yd(x) = −yg(x)d(x) for all x, y ∈ N. (46)
Replacing y by yt in (46) and using (46) again, we have
g(x)ytd(x) = −ytg(x)d(x)
= (−y)(tg(x)d(x))
= (−y)(−g(x)td(x))
= (−y)(−g(x))td(x) for all x, y, t ∈ N.
But since g is an additive map, we have −g(x) = g(−x) and g(x)ytd(x) =
(−y)g(−x)td(x) for all x, y, t ∈ N , which can be rewritten as
(−g(−x)y + yg(−x))Nd(x) = {0} for all x, y ∈ N.
Taking −x instead of x in the latter relation and using the 3-primeness of
N , we arrive at
g(x) ∈ Z(N) or d(x) = 0 for all x ∈ N.
To complete the proof, we only need to consider the same arguments as
used after (39) in the proof of Theorem 6, we arrive at a conclusion N is
A. Raji 297
a commutative ring. Now, returning to the assumptions of theorem, we
obtain
F (xy) = F (x)y + g(x)d(y) = xy for all x, y ∈ N.
Putting xz instead of x, we arrive at
g(x)g(z)d(y) = 0 for all x, y, z ∈ N. (47)
Taking into account that g is onto, (47) yields
xNd(y) = {0} for all x, y ∈ N.
In the light of the 3-primeness of N , the laste equation forces that d = 0,
which is a contradiction.
The following example shows that the condition of 3-primeness in the
hypothesis of Theorems 4, 5, 6 and 7 is crucial.
Example 2. Let S be a noncommutative 2-torsion free zero-symmetric
right near-ring. Let us deőne N,F, d, g : N → N by:
N =
{(
0 x y
0 0 0
0 0 0
)
| 0, x, y ∈ S
}
, F
(
0 x y
0 0 0
0 0 0
)
=
(
0 x 0
0 0 0
0 0 0
)
,
d
(
0 x y
0 0 0
0 0 0
)
=
(
0 y x
0 0 0
0 0 0
)
and g = d.
It can be checked that N with matrix addition and matrix multiplication is
not a 3-prime right near-ring and F is a nonzero generalized semiderivation
of N associated with a nonzero semiderivation d which is associated with
the automorphism map g such that the following situations hold:
(i) F ([A,B]) = 0, (ii) F (A◦B) = 0, (iii) F ([A,B]) = [A,B], (iv) F (A◦B) =
A ◦B for all A,B ∈ N . However, N is not a commutative ring.
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298 Commutativity criteria for 3-prime near-rings
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Contact information
Abderrahmane Raji LMACS Laboratory
Faculty of Sciences and Technology
Sultan Moulay Slimane University
P.O.Box 523, 23000, Beni Mellal, Morocco
E-Mail(s): rajiabd2@gmail.com
Received by the editors: 18.08.2019
and in őnal form 26.04.2021.
mailto:rajiabd2@gmail.com
A. Raji
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