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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 38, Num. 1 (2023), 51 – 66

doi:10.17398/2605-5686.38.1.51

Available online January 23, 2023

On Jordan ideals with left derivations
in 3-prime near-rings

A. En-guady, A. Boua@

Department of Mathematics, Polydisciplinary Faculty of Taza
Sidi Mohammed Ben Abdellah University, Fez, Morocco

adel.enguady@usmba.ac.ma , abdelkarimboua@yahoo.fr

Received September 12, 2022 Presented by C. Mart́ınez
Accepted December 13, 2022

Abstract: We will extend in this paper some results about commutativity of Jordan ideals proved

in [2] and [6]. However, we will consider left derivations instead of derivations, which is enough to
get good results in relation to the structure of near-rings. We will also show that the conditions

imposed in the paper cannot be removed.

Key words: 3-prime near-rings, Jordan ideals, Left derivations.

MSC (2020): 16N60; 16W25; 16Y30.

1. Introduction

A right (resp. left) near-ring A is a triple (A, +, .) with two binary
operations ” + ” and ”.” such that:

(i) (A, +) is a group (not necessarily abelian),
(ii) (A, .) is a semigroup,
(iii) (r + s).t = r.t + s.t (resp. r.(s + t) = r.s + r.t) for all r; s; t ∈A.

We denote by Z(A) the multiplicative center of A, and usually A will be
3-prime, that is, for r,s ∈ A, rAs = {0} implies r = 0 or s = 0. A right
(resp. left) near-ring A is a zero symmetric if r.0 = 0 (resp. 0.r = 0) for
all r ∈ A, (recall that right distributive yields 0r = 0 and left distributive
yields r.0 = 0). For any pair of elements r,s ∈ A, [r,s] = rs − sr and
r◦s = rs + sr stand for Lie product and Jordan product respectively. Recall
that A is called 2-torsion free if 2r = 0 implies r = 0 for all r ∈ A. An
additive subgroup J of A is said to be Jordan left (resp. right) ideal of A if
r ◦ i ∈ J (resp. i ◦ r ∈ J) for all i ∈ J, r ∈ A and J is said to be a Jordan
ideal of A if r◦ i ∈ J and i◦r ∈ J for all i ∈ J, r ∈N . An additive mapping

@ Corresponding author

ISSN: 0213-8743 (print), 2605-5686 (online)

c© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.38.1.51
mailto:adel.enguady@usmba.ac.ma
mailto:abdelkarimboua@yahoo.fr
mailto:abdelkarimboua@yahoo.fr
https://publicaciones.unex.es/index.php/EM
https://creativecommons.org/licenses/by-nc/3.0/


52 a. en-guady, a. boua

H : A → A is a multiplier if H(rs) = rH(s) = H(r)s for all r,s ∈ A. An
additive mapping d : A→A is a left derivation (resp. Jordan left derivation)
if d(rs) = rd(s) + sd(r) (resp. d(r2) = 2rd(r)) holds for all r,s ∈ A. The
concepts of left derivations and Jordan left derivations were introduced by
Breşar et al. in [7], and it was shown that if a prime ring R of characteristic
different from 2 and 3 admits a nonzero Jordan left derivation, then R must
be commutative. Obviously, every left derivation is a Jordan left derivation,
but the converse need not be true in general (see [9, Example 1.1]). In [1],
M. Ashraf et al. proved that the converse statement is true in the case when
the underlying ring is prime and 2-torsion free. The study of left derivation
was developed by S.M.A. Zaidi et al. in [9] and they showed that if J is a
Jordan ideal and a subring of a 2-torsion-free prime ring R admits a nonzero
Jordan left derivation and an automorphism T such that d(r2) = 2T(r)d(r)
holds for all r ∈ J, then either J ⊆ Z(R) or d(J) = {0}. Recently, there
have been many works concerning the Jordan ideals of near-rings involving
derivations; see, for example, [4], [5], [6], etc. For more details, in [6, Theorem
3.6 and Theorem 3.12], we only manage to show the commutativity of the
Jordan ideal, but we don’t manage to show the commutativity of our studied
near-rings, hence our goal to extend these results to the left derivations.

2. Some preliminaries

To facilitate the proof of our main results, the following lemmas are
essential.

Lemma 2.1. Let N be a 3-prime near-ring.

(i) [3, Lemma 1.2 (iii)] If z ∈ Z(N) \ {0} and xz ∈ Z(N) or zx ∈ Z(N),
then x ∈ Z(N).

(ii) [2, Lemma 3 (ii)] If Z(N) contains a nonzero element z of N which
z + z ∈ Z(N), then (N , +) is abelian.

(iii) [5, Lemma 3] If J ⊆ Z(N), then N is a commutative ring.

Lemma 2.2. ([8, Theorem 3.1]) Let N be a 3-prime right near-ring. If
N admits a nonzero left derivation d, then the following properties hold true:

(i) If there exists a nonzero element a such that d(a) = 0, then a ∈ Z(N),

(ii) (N , +) is abelian, if and only if N is a commutative ring.



jordan ideals with left derivations 53

Lemma 2.3. ([4, Lemma 2.2]) Let N be a 3-prime near-ring. If N
admits a nonzero Jordan ideal J, then j2 6= 0 for all j ∈ J \{0}.

Lemma 2.4. ([4, Theorem 3.1]) Let N be a 2-torsion free 3-prime right
near-ring and J a nonzero Jordan ideal of N . If N admits a nonzero left
multiplier H, then the following assertions are equivalent:

(i) H(J) ⊆ Z(N);

(ii) H(J2) ⊆ Z(N);

(iii) N is a commutative ring.

Lemma 2.5. ([5, Theorem 1]) Let N be a 2-torsion free 3-prime near-
ring and J a nonzero Jordan ideal of N . Then N must be a commutative ring
if J satisfies one of the following conditions:

(i) i◦ j ∈ Z(N) for all i,j ∈ J.

(ii) i◦ j ± [i,j] ∈ Z(N) for all i,j ∈ J.

Lemma 2.6. Let N be a left near-ring. If N admits a left derivation d,
then we have the following identity:

xyd(yn) = yxd(yn) for all n ∈ N, x,y ∈N .

Proof. Using the definition of d. On one hand, we have

d(xyn+1) = xd(yn+1) + yn+1d(x)

= xynd(y) + xyd(yn) + yn+1d(x) for all n ∈ N, x,y ∈N .

On the other hand

d(xyn+1) = xynd(y) + yd(xyn)

= xynd(y) + yxd(yn) + yn+1d(x) for all n ∈ N, x,y ∈N .

Comparing the two expressions, we obtain the required result.



54 a. en-guady, a. boua

3. Results characterizing left derivations in 3-prime near-rings

In [2], the author proved that if N is a 3-prime 2-torsion-free near-ring
which admits a nonzero derivation D for which D(N) ⊆ Z(N), then N is
a commutative ring. In this section, we investigate possible analogs of these
results, where D is replaced by a left derivation d and by integrating Jordan
ideals.

Theorem 3.1. Let N be a 2-torsion free 3-prime near-ring and J be a
nonzero Jordan ideal of N . If N admits a left derivation d, then the following
assertions are equivalent:

(i) d(J) ⊆ Z(N);
(ii) d(J2) ⊆ Z(N);
(iii) N is a commutative ring or d = 0.

Proof. Case 1: N is a 3-prime right near-ring. It is obvious that (iii)
implies (i) and (ii). Therefore we only need to prove (i)⇒(iii) and (ii)⇒(iii).

(i)⇒(iii): Suppose that Z(N) = {0}, then d(J) = {0}. From Lemma
2.2 (i), we get J ⊆ Z(N) and by Lemma 2.1 (i), we conclude that N is a
commutative ring. In this case, and by using the definition of d together with
the 2-torsion freeness of N , the above equation leads to

jd(n) = 0 for all j ∈ J, n ∈N . (3.1)

Taking j◦m of j, where m ∈N in (3.1) and using it, we get JNd(n) = {0}
for all n ∈N . Since N is 3-prime and J 6= {0}, then d = 0.

Now suppose Z(N) 6= {0}. By assumption, we have d(j ◦ j) ∈ Z(N) for
all j ∈ J, which gives (4j)d(j) ∈ Z(N) for all j ∈ J, that is (d(4j))j ∈ Z(N)
for all j ∈ J. Invoking Lemma 2.1 (i) and Lemma 2.2 (i) together with the
2-torsion freeness of N , we obtain J ⊆ Z(N), and Lemma 2.4 (i) forces that
N is a commutative ring.

(ii)⇒(iii): Suppose that Z(N) = {0}, then d(J2) = {0}, which implies
J2 ⊆ Z(N) by Lemma 2.2 (ii), hence N is a commutative ring by Lemma
2.4 (ii). Now using assumption, then we have d(j2) = 0 for all j ∈ J. By
the 2-torsion freeness of N , it follows jd(j) = 0 for all j ∈ J. Since N is a
commutative ring, we can write jnd(j) = 0 for all j ∈ J, n ∈N , which implies
that jNd(j) = {0} for all j ∈ J. By the 3-primeness of N , we conclude that
d(J) = {0}. Using the same techniques as we have used in the proof of
(i)⇒(iii) one can easily see that d = 0 .



jordan ideals with left derivations 55

Now suppose Z(N) 6= {0}. By our hypothesis, we have d((j◦j2)j) ∈ Z(N)
for all j ∈ J, and by a simplification, we find d((j2 ◦ j)j) = (j2)d(4j2) for all
j ∈ J :

d((j2 ◦ j)j) = d((j3 + j3)j) = d(j4 + j4) = d(2j2j2)
= 2j2d(j2) + j2d(2j2) = 2j2d(j2) + d(2j2)j2

= 2j2d(j2) + 2j2d(j2) = 4j2d(j2) = j2d(4j2).

Hence, j2d(4j2) ∈ Z(N) for all j ∈ J, which implies j2d((4j)(j)) ∈ Z(N)
for all j ∈ J. Invoking Lemma 2.1 (i), then j2 ∈ Z(N) or 4d(j2) = 0 for all
j ∈ J. In view of the 2-torsion freeness of N together with Lemma 2.2 (i), we
can assure that

j2 ∈ Z(N) for all j ∈ J. (3.2)
Applying the definition of d together with our hypothesis, and (3.2), we

have for all j ∈ J and x ∈N :

d(xj4) = d(xj2j2) = xj2d(j2) + j2d(xj2)

= xj2d(j2) + d(xj2)j2 = xj2d(j2) + xj2d(j2) + j4d(x)

= j2d(j2)x + j2d(j2)x + j4d(x) = (2j2d(j2))x + j4d(x) ,

d(xj4) = xd(j4) + j4d(x) = x(2j2d(j2)) + j4d(x) .

Comparing the two expressions, we obtain

x(2j2d(j2)) = (2j2d(j2))x for all j ∈ J, x ∈N .

Consequently, 2j2d(j2) ∈ Z(N) for all j ∈ J. According to Lemma 2.1 (i)
and Lemma 2.2 (i), that follows 2j2 ∈ Z(N) for all j ∈ J, which implies
(N , +) is abelian by Lemma 2.1 (ii), and Lemma 2.2 (ii) assures that N is a
commutative ring.

Case 2: N is a 3-prime left near-ring. It is obvious that (iii) implies (i)
and (ii).

(i)⇒(iii): Suppose that Z(N) = {0}. Using our hypothesis, then we have
d(j ◦ n) = 0 for all j ∈ J, n ∈ N . Applying definition of d and using our
assumption with the 2-torsion freeness of N , we get

jd(n) = 0 for all n ∈N . (3.3)

Replacing n by jnm in (3.3) and using it, then we get j2nd(m) = 0 for all
j ∈ J, n,m ∈ N , which implies that j2Nd(m) = {0} for all j ∈ J, m ∈ N .
Using Lemma 2.3 together with the 3-primeness of N , it follows that d = 0.



56 a. en-guady, a. boua

Now assuming that Z(N) 6= {0}. By Lemma 2.6, we can write jnd(j) =
njd(j) for all j ∈ J, n ∈N , which reduces to d(j)N [j,m] = {0} for all j ∈ J,
m ∈N and by the 3-primeness of N , we conclude that

j ∈ Z(N) or d(j) = 0 for all j ∈ J. (3.4)

Suppose that there is j0 ∈ J such that d(j0) = 0. Using our hypothesis,
then we have d(j0(j0 ◦n)) ∈ Z(N) for all n ∈ N . Applying the definition of
d and using our assumption, we get j0d((j0 ◦n)) ∈ Z(N) for all n ∈ N . By
Lemma 2.1 (i), we conclude

j0 ∈ Z(N) or d((j0 ◦n)) = 0 for all n ∈N . (3.5)

If d((j0 ◦n)) = 0 for all n ∈N , using the 2-torsion freeness of N , we get

j0d(n) = 0 for all n ∈N . (3.6)

Replacing n by j0nm in (3.6) and using it, then we get j
2
0nd(m) = 0 for

all n,m ∈ N . Since d 6= 0, the 3-primeness of N gives j20 = 0, which is a
contradiction with Lemma 2.3. Then (3.4) becomes J ⊆ Z(N), which forces
that N is commutative ring by Lemma 2.1 (iii).

(ii)⇒(iii): Suppose that Z(N) = {0}, then d(j2) = 0 for all j ∈ J, by
the 2-torsion freeness of N , we get

jd(j) = 0 for all j ∈ J. (3.7)

Using Lemma 2.6, we can write jnd(j) = njd(j) for all j ∈ J, n ∈ N ,
from (3.7), we get jnd(j) = 0 for all j ∈ J, n ∈ N , which implies jNd(j) =
{0} for all j ∈ J, n ∈ N and by the 3-primeness of N , we deduce that
d(J) = {0}. Using the same techniques as used in the proof of (i)⇒(iii), we
conclude that d = 0.

Assuming that Z(N) 6= {0}. By Lemma 2.5, we can write

jnd(j2) = njd(j2) for all x,y ∈N , (3.8)

which implies that

d(j2)N [j,m] = {0} for all j ∈ J, m ∈N .

By the 3-primeness of N , we conclude that

j ∈ Z(N) or d(j2) = 0 for all j ∈ J. (3.9)



jordan ideals with left derivations 57

If there exists j0 ∈ J such that d(j20 ) = 0, using the definition of d and the
2-torsion freeness of N , then we have

j0d(j0) = 0. (3.10)

By Lemma 2.6, we can write j0Nd(j0) = {0}. In view of the 3-primeness of
N , that follows d(j0) = 0. Using our hypothesis, we have d(j0(2i2)) ∈ Z(N)
for all i ∈ J. Applying the definition of d and using our assumption, we get
j0d(2i

2) ∈ Z(N) for all i ∈ J. By the 2-torsion freeness of N and Lemma
2.1 (i) we conclude

j0 ∈ Z(N) or id(i) = 0 for all i ∈ J. (3.11)

If id(i) = 0 for all i ∈ J. Using the same techniques as used in the proof
of (ii)⇒(iii), we conclude that d = 0. Then (3.9) becomes

J ⊆ Z(N) or d = 0.

Corollary 3.2. Let N be a 2-torsion free 3-prime near-ring. If N admits
a left derivation d, then the following assertions are equivalent:

(i) d(N) ⊆ Z(N);
(ii) d(N2) ⊆ Z(N);
(iii) N is a commutative ring or d = 0.

The following example proves that the 3-primeness of N in Theorem 3.1
cannot be omitted.

Example 3.3. Let R be a 2-torsion right or left near-ring which is not
abelian. Define N , J and d by:

N =




0 0 0r 0 0
s t 0


 : r,s,t, 0 ∈R


 ,J =




0 0 00 0 0
p 0 0


 : p, 0 ∈R


 ,

d


0 0 0r 0 0
s t 0


 =


0 0 00 0 0

0 t 0


 .

Then N is a right or left near-ring which is not 3-prime, J is a nonzero
Jordan ideal of N and d is a nonzero left derivation of N which is not a
derivation. It is easy to see that



58 a. en-guady, a. boua

(i) d(J) ⊆ Z(N).
(ii) d(J2) ⊆ Z(N).

However, neither d = 0 nor N is a commutative ring.

4. Some polynomial identities in right near-rings
involving left derivations

This section is motivated by [6, Theorem 3.6 and Theorem 3.12]. Our aim
in the current paper is to extend these results of Jordan ideals on 3-prime
near-rings admitting a nonzero left derivation.

Theorem 4.1. Let N be a 2-torsion free 3-prime near-ring and J be a
nonzero Jordan ideal of N . If N admits a nonzero left derivation d and a
multiplier H satisfying d(x◦ j) = H(x◦ j) for all j ∈ J, x ∈ N , then N is a
commutative ring.

Proof. Assume that d(x◦j) = H(x◦j) for all j ∈ J, x ∈N . If H = 0, the
last equation becomes d(x◦j) = 0 for all j ∈ J, x ∈N . And recalling Lemma
2.2 (ii), then (x◦j) ∈ Z(N) for all j ∈ J, x ∈N , so N is a commutative ring
by Lemma 2.5 (i).

Now assume that H 6= 0 and d(x ◦ j) = H(x ◦ j) for all j ∈ J, x ∈ N .
Replacing x by xj and using the fact that (xj ◦ j) = (x◦ j)j, we get

d((x◦ j)j) = H((x◦ j)j) for all i,j ∈ J, x ∈N .

By the definition of d and H, we obtain

(x◦ j)d(j) + jd(x◦ j) = H(x◦ j)j for all i,j ∈ J, x ∈N .

Replacing j by (y ◦ i), where i ∈ J, y ∈ N , in the preceding expression,
we can see that

(x◦ (y ◦ i))d((y ◦ i)) + (y ◦ i)d(x◦ (y ◦ i)) = H(x◦ (y ◦ i))(y ◦ i)

for all, i,j ∈ J, x,y ∈N .
By a simplification, we thereby obtain

(y ◦ i)H(x◦ (y ◦ i)) = 0 for all i,j ∈ J, x,y ∈N . (4.1)

Applying H on (4.1), it follows that

(y ◦ i)H(H(x◦ (y ◦ i))) = 0 for all i,j ∈ J, x,y ∈N . (4.2)



jordan ideals with left derivations 59

Applying d on (4.1) and recalling (4.2), we get

H(x◦ (y ◦ i))H(y ◦ i) = 0 for all x,y ∈N , (4.3)

which gives

xH(y ◦ i)H(y ◦ i) = −H(y ◦ i)xH(y ◦ i) for all x,y ∈N .

Substituting xz instead of x in preceding equation and applying it, we
obviously obtain

xzH(y ◦ i)H(y ◦ i) = (−H(y ◦ i))xzH(y ◦ i)
= x(−H(y ◦ i))zH(y ◦ i) for all x,y,z ∈N .

This forces that

[x, (−H(y ◦ i))]zH(y ◦ i) = 0 for all x,y,z ∈N .

Then [x, (−H(y◦i))]NH(y◦i) = {0} for all x,y ∈N . By the 3-primeness
of N , we get

(−H(y ◦ i)) ∈ Z(N) for all i ∈ J, y ∈N . (4.4)

Substituting yi instead y in (4.4), (−H(y◦i))i ∈ Z(N) for all i ∈ J, y ∈N .
It follows that Lemma 2.1 (i)

H(y ◦ i) = 0 or i ∈ Z(N) for all i ∈ J, y ∈N . (4.5)

Suppose that there exists an element i0 ∈ J such that

H(y ◦ i0) = 0 for all y ∈N , (4.6)

which implies (−i0)H(y) = H(y)i0 for all y ∈ N . Replacing y by xyz in the
last equation, we get

(−i0)H(xyz) = H(xyz)i0 for all x,y,z ∈N ,

which means that

(−i0)xyH(z) = x(−i0)yH(z) for all x,y,z ∈N ,

so [x,−i0]NH(z) = {0} for all x,z ∈ N . Since H 6= 0 and N is 3-prime,
we get −i0 ∈ Z(N). Now substituting −i0 instead i in (4.4), we obtain



60 a. en-guady, a. boua

−H(y ◦ (−i0)) ∈ Z(N) for all y ∈ N , which implies (−H(2y))(−i0) ∈ Z(N)
for all y ∈ N , using Lemma 2.1 (i), we get −2H(y) ∈ Z(N) for all y ∈ N or
i0 = 0. Thus (4.5) becomes

− 2H(y) ∈ Z(N) for all y ∈N or J ⊆ Z(N). (4.7)

Case 1: If −2H(y) ∈ Z(N) for all y ∈ N . Replacing y by zt in the last
equation, we obtain (−2H(z))t ∈ Z(N) for all z,t ∈N . Since N is 2-torsion
free and H 6= 0, we obtain N ⊆ Z(N) by Lemma 2.1 (ii). Which assures that
N is a commutative ring by Lemma 2.1 (iii).

Case 2: If J ⊆ Z(N), then N is a commutative ring by virtue of
Lemma 2.1 (iii).

The next result is an immediate consequence of Theorem 3.1, just to take
H = idN in Theorem 4.1.

Corollary 4.2. Let N be a 2-torsion free 3-prime near-ring and J be a
nonzero Jordan ideal of N . If N admits a nonzero left derivation d such that
d(x◦ j) = x◦ j for all j ∈ J, x ∈N , then N is a commutative ring.

Theorem 4.3. Let N be a 2-torsion free 3-prime near-ring and J be a
nonzero right Jordan ideal of N . If N admits a left derivation d and a nonzero
multiplier H satisfying any one of the following identities:

(i) d(H(J)) = {0};
(ii) d(H(J2)) = {0};
(iii) d(H(n◦ j)) = d(H([n,j])) for all j ∈ J, n ∈N ;
(iv) d(H(nj)) = H(j)d(n) for all j ∈ J, n ∈N ,

then d = 0.

Proof. (i) Assume that d (H(J)) = {0}. Therefore, by Lemma 2.2 (i) and
Lemma 2.4 (i), N is a commutative ring. Using our hypothesis and by the
2-torsion freeness of N , we can see d(H(j)n) = 0 for all j ∈ J, n ∈ N .
Applying the definition of d, we obtain

H(j)d(n) = 0 for all j ∈ J, n ∈N . (4.8)

Replacing j by j◦m, where m ∈N in (4.8) and using it, we can easily arrive
at H(J)Nd(n) = {0} for all n ∈ N . By the 3-primeness of N , we conclude



jordan ideals with left derivations 61

that d(N) = {0} or H(J) = {0}. If H(J) = {0}, then H((j◦m)◦n)) = 0 for all
j ∈ J, n,m ∈N . In view of the 2-torsion freeness of N , we get JNH(n) = {0}
and by the 3-primeness of N , we obtain J = {0} or H(n) = {0}, that would
contradict with our hypothesis, then d = 0.

(ii) Suppose that d
(
H(J2)

)
= {0}, according to Lemma 2.2 (i) and Lemma

2.4 (i), N is a commutative ring. Now using our hypothesis, d(H(i(j◦n))) = 0
for all i,j ∈ J, n ∈N , by the 2-torsion freeness of N , we can see d(H(ijn)) = 0
for all i,j ∈ J, n ∈N . Applying the definition of d, we obtain

iH(j)d(n) = 0 for all i,j ∈ J, n ∈N . (4.9)

Substituting j◦m for j, where m ∈N and i◦t for j, where t ∈N in (4.9)
and using it, we can easily arrive at JNH(J)Nd(n) = {0} for all n ∈ N .
By the 3-primeness of N , we conclude that d(N) = {0} or H(J) = {0} or
J = {0}. If H(J) = {0}, using the same techniques as we have used in the
proof of (i), one can easily find d = 0.

(iii) Suppose that d(H(n◦j)) = d(H([n,j])) for all j ∈ J, n ∈N . Taking
nj instead of n, we obtain

d(H((n◦ j)j)) = d(H([n,j]j)) for all j ∈ J, n ∈N .

Using the definition of d, we get

H(n◦ j)d(j) + jd(H(n◦ j)) = H([n,j])d(j) + jd(H([n,j]))

for all j ∈ J, n ∈N .
By a simplification, we can rewrite this equation as

2jH(n)d(j) = 0 for all j ∈ J, n ∈N .

Substituting zyt for n, where x,y,z ∈N in last equation, we can see

2jyH(z)td(j) = 0 for all j ∈ J, y,z,t ∈N .

By the 2-torsion freeness of N , the above equation becomes jNH(z)Nd(j)
= {0} for all j ∈ J,z ∈ N . Since N is 3-prime and H 6= 0, it follows that
d(J) = {0}, which forces that d = 0 by (i).

(iv) Suppose that d(H(nj)) = H(j)d(n) for all j ∈ J, n ∈ N . From this
equation we obtain

d(nH(j)) = H(j)d(n) for all j ∈ J, n ∈N .



62 a. en-guady, a. boua

Using the definition of d, we have

nd(H(j)) + H(j)d(n) = H(j)d(n) for all j ∈ J, n ∈N .

Then nd(H(j)) = 0 for all j ∈ J, n ∈N , which implies that d(H(J)) = {0}
by invoking the 3-primeness of N , and consequently d = 0 by (i).

The next result is an immediate consequence of Theorem 3.1, just to take
H = idN in Theorem 4.6.

Corollary 4.4. Let N be a 2-torsion free 3-prime near-ring and J be a
nonzero right Jordan ideal of N . If N admits a left derivation d and a nonzero
multiplier H satisfying any one of the following identities:

(i) d(J) = {0};
(ii) d(J2) = {0};
(iii) d(n◦ j) = d([n,j]) for all j ∈ J, n ∈N ,
(iv) d(nj) = jd(n) for all j ∈ J, n ∈N ;

then d = 0.

The following example proves that the 3-primeness of N in Theorem 4.1
and Theorem 4.3 cannot be omitted.

Example 4.5. Let S be a 2-torsion right near ring which is not abelian.
Define N , J, d and H by:

N =




0 0 p0 q 0

0 0 0


 : p,q, 0 ∈S


 , J =




0 0 00 s 0

0 0 0


 : s, 0 ∈S


 ,

d


0 0 p0 q 0

0 0 0


 =


0 0 p0 0 0

0 0 0


 and H


0 0 p0 q 0

0 0 0


 =


0 0 00 q 0

0 0 0


 .

Then N is a right near-ring which is not 3-prime, J is a nonzero Jordan
ideal of N , d is a nonzero left derivation of N , and H is a nonzero multiplier
of N , such that

(i) d(x◦ j) = H(x◦ j) for all j ∈ J, x ∈N ;
(ii) d (H(J)) = {0};



jordan ideals with left derivations 63

(iii) d
(
H(J2)

)
= {0};

(iv) d(H(n◦ j)) = d(H([n,j])) for all j ∈ J, n ∈N ;
(v) d(H(nj)) = H(j)d(n) for all j ∈ J, n ∈N .

However, neither d = 0 nor N is a commutative ring.

Theorem 4.6. Let N be a 2-torsion free 3-prime near-ring and J be a
nonzero Jordan ideal of N and let H a nonzero multiplier on N . Then there
is no nonzero left derivation d such that d(x ◦ j) = H([x,j]) for all j ∈ J,
x ∈N .

Proof. Assume that

d(x◦ j) = H([x,j]) for all j ∈ J, x ∈N . (4.10)

Replacing x by j, in (4.10), we get

2d(j2) = d(j2 + j2) = d(j ◦ j) = 0 for all j ∈ J.

By the 2-torsion freeness of N , we get

0 = d(j2) = 2jd(j) for all j ∈ J. (4.11)

In view of the 2-torsion freeness of N , this easily yields

jd(j) = 0 for all j ∈ J. (4.12)

Replacing x by xj in (4.10), we get

d(xj ◦ j) = H([xj,j]) for all j ∈ J, x ∈N .

Using the fact that (xj ◦ j) = (x◦ j)j and [xj,j] = [x,j]j, we obtain

d((x◦ j)j) = H([x,j]j) for all j ∈ J, x ∈N .

By the definition of d, the last equation is expressible as

(x◦ j)d(j) = [H([x,j]),j] for all j ∈ J, x ∈N .

Substituting xj instead x, it follows from (4.12) that

[H([xj,j]),j] = 0 for all j ∈ J, x ∈N . (4.13)



64 a. en-guady, a. boua

Replacing x by d(j)x in (4.13) and using (4.12), we can easily arrive at

[d(j)H(x)j2,j] = 0 for all j ∈ J, x ∈N .

Which reduces to

d(j)H(x)j3 = 0 for all j ∈ J, x ∈N .

Substituting rst instead x where r,s,t ∈ N in the last equation, we get
d(j)rH(s)tj3 = 0 for all j ∈ J, r,s,t ∈ N , which implies d(j)NH(s)Nj3 =
{0} for all j ∈ J, s ∈N . Since H 6= 0 and using the 3-primeness hypothesis,
it follows that

d(j) = 0 or j3 = 0 for all j ∈ J. (4.14)

Suppose that there exists an element j0 ∈ J \ {0} such that j30 = 0.
Replacing j by j0 and x by xj

2
0 in (4.10) and using (4.12), then

d(xj20 ◦ j0) = H([xj
2
0,j0] for all x ∈N .

Using our assumption, we find that

d(j0xj
2
0 ) = H(−j0xj

2
0 ) for all x ∈N .

By the definition of d, we get

j0d(xj
2
0 ) + xj

2
0d(j0) = −j0H(x)j

2
0 for all x ∈N .

In light of equation (4.12), it follows easily that

j0d(xj
2
0 ) = −j0H(x)j

2
0 for all x ∈N .

So, by (4.14) and (4.12), we get

−j0H(x)j20 = 0 for all x ∈N .

Substituting rst instead x gives −j0rH(s)tj20 = 0 for all r,s,t ∈N , which
implies (−j0)NH(s)Nj20 = {0} for all s ∈N . Since H 6= 0, by the 3-primeness
of N and Lemma 2.3, the preceding expression leads to j0 = 0.

Hence, (4.14) becomes d(J) = {0}, which leads to d = 0 by Theorem
3.1 (i); a contradiction.

Corollary 4.7. Let N be a 2-torsion free 3-prime near-ring and J be a
nonzero Jordan ideal of N . Then there is no nonzero left derivation d such
that d(x◦ j) = [x,j] for all j ∈ J, x ∈N .



jordan ideals with left derivations 65

Theorem 4.8. Let N be a 2-torsion free 3-prime near-ring and J be a
nonzero Jordan ideal of N . Then N admits no nonzero left derivation d such
that d([x,j]) = d(x)j for all j ∈ J, x ∈N .

Proof. Assume that

d([x,j]) = d(x)j for all x ∈N , j ∈ J. (4.15)

Replacing x by j in (4.15), we get

d(j)j = 0 for all j ∈ J. (4.16)

Substituting xj instead of x in (4.15), we obtain

d([xj,j]) = d(xj)j for all j ∈ J, x ∈N .

Notice that [xj,j] = [x,j]j, the last relation can be rewritten as

d([x,j]j) = (xd(j) + jd(x))j for all j ∈ J, x ∈N .

The definition of d gives us

[x,j])d(j) + jd([x,j]) = jd(x)j for all j ∈ J, x ∈N .

Using our assumption, we obviously obtain

xjd(j) = jxd(j) for all j ∈ J, x ∈N . (4.17)

Replacing x by yt in (4.17) and invoking it, we can see that

yjtd(j) = jytd(j) for all j ∈ J, y,t ∈N .

The last equation gives us [y,j]Nd(j) = {0} for all j ∈ J, x ∈N . By the
3-primeness of N , we get

j ∈ Z(N) or d(j) = 0 for all j ∈ J. (4.18)

If there exists j0 ∈ J such that d(j0) = 0. Using Lemma 2.4, we obtain
j0 ∈ Z(N). In this case, (4.18) becomes J ⊆ Z(N) which forces that N is a
commutative ring by Lemma 2.1 (i). Hence (4.6) implies that d(x)j = 0 for
all j ∈ J, x ∈N . Replacing j by j ◦ t in the last equation, it is obvious that
2d(x)tj = 0 for all j ∈ J, t,x ∈ N . It follows from the 2-torsion freeness of
N that d(x)Nj = {0} for all j ∈ J, x ∈ N . By the 3-primeness of N , we
conclude that d = 0 or J = {0}; a contradiction.



66 a. en-guady, a. boua

5. Conclusion

In this paper, we study the 3-prime near-rings with left derivations. We
prove that a 3-prime near-ring that admits a left derivation satisfying cer-
tain differential identities on Jordan ideals becomes a commutative ring. In
comparison to some recent studies that used derivations, these results are
considered more developed. In future research, one can discuss the following
issues:

(i) Theorem 3.1, Theorem 4.1, Theorem 4.3 and Theorem 4.6 can be proven
by replacing left derivation d by a generalized left derivation.

(ii) The study of 3-prime near-rings that admit generalized left derivations
satisfying certain differential identities on Lie ideals is another interest-
ing work for the future.

References

[1] M. Ashraf, N. Rehman, On Lie ideals and Jordan left derivation of prime
rings, Arch. Math. (Brno) 36 (2000), 201 – 206.

[2] H.E. Bell, G. Mason, On derivations in near-rings, in “Near-rings and
near-fields”, North Holland Math. Stud. 137, North-Holland, Amsterdam,
1987, 31 – 35.

[3] H.E. Bell, On derivations in near-rings II, in “Nearrings, nearfields and K-
loops”, Math. Appl. 426, Kluwer Acad. Publ., Dordrecht, 1997, 191 – 197.

[4] A. Boua, H.E. Bell, Jordan ideals and derivations satisfying algebraic iden-
tities, Bull. Iranian Math. Soc. 44 (2018), 1543 – 1554.

[5] A. Boua, L. Oukhtitei, A. Raji, Jordan ideals and derivations in prime
near-rings, Comment. Math. Univ. Carolin. 55 (2) (2014), 131 – 139.

[6] A. Boua, Commutativity of near-rings with certain constrains on Jordan ide-
als, Bol. Soc. Parana. Mat. (3) 36 (4) (2018), 159 – 170.

[7] M. Brešar, J. Vukman, On left derivations and related mappings, Proc.
Amer. Math. Soc. 110 (1) (1990), 7 – 16.

[8] A. Enguady, A. Boua, On Lie ideals with left derivations in 3-prime near-
rings, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 68 (1) (2022), 123 – 132.

[9] S.M.A. Zaidi, M. Ashraf, A. Shakir, On Jordan ideals and left
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1957 – 1964.


	Introduction
	Some preliminaries
	Results characterizing left derivations in 3-prime near-rings
	Some polynomial identities in right near-rings involving left derivations
	Conclusion