Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings
We give some classes of zero-symmetric 3-prime near-rings such that every member of these classes has no nonzero derivation. Moreover, we extend the concept of “3-prime” to subsets of near-rings and use it to generalize Theorem 1.1 due to Fong, Ke, and Wang concerning the transformation near-rings M...
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|---|---|
| author | Al-Shaalan, Khalid H. Kamal, Ahmed A. M. Аль-Шаалан, Халід Г. Камаль, Ахмед А. М. |
| author_facet | Al-Shaalan, Khalid H. Kamal, Ahmed A. M. Аль-Шаалан, Халід Г. Камаль, Ахмед А. М. |
| author_sort | Al-Shaalan, Khalid H. |
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| collection | OJS |
| datestamp_date | 2019-12-05T10:24:58Z |
| description | We give some classes of zero-symmetric 3-prime near-rings such that every member of these classes has no nonzero derivation. Moreover, we extend the concept of “3-prime” to subsets of near-rings and use it to generalize Theorem 1.1 due to Fong, Ke, and Wang concerning the transformation near-rings M o (G) by using a different technique and a simpler proof. |
| first_indexed | 2026-03-24T02:19:32Z |
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| fulltext |
UDC 512.5
Ahmed A. M. Kamal (College Sci., King Saud Univ., Kingdom of Saudi Arabia),
Khalid H. Al-Shaalan (King Abdul-aziz Military Academy, Kingdom of Saudi Arabia)
NONEXISTENCE OF NONZERO DERIVATIONS ON SOME CLASSES
OF ZERO-SYMMETRIC 3-PRIME NEAR-RINGS
НЕIСНУВАННЯ НЕНУЛЬОВИХ ПОХIДНИХ НА ДЕЯКИХ КЛАСАХ
3-ПРОСТИХ МАЙЖЕ-КIЛЕЦЬ З НУЛЬОВОЮ СИМЕТРIЄЮ
We give some classes of zero-symmetric 3-prime near-rings such that every member in these classes has no nonzero
derivation. Moreover, we extend the concept of ”3-prime” to subsets of near-rings and use it to generalize Theorem 1.1 due
to Fong, Ke, and Wang concerning the transformation near-rings Mo(G) by using a different technique and a more simple
proof.
Наведено деякi класи 3-простих майже-кiлець з нульовою симетрiєю таких, що будь-який елемент цих класiв не має
ненульової похiдної. Крiм того, поняття „3-простих“ узагальнено на пiдмножини майже-кiлець i застосовано, щоб
узагальнити теорему 1.1 Фонга, Ке i Ванга про трансформацiю майже-кiлець Mo(G) за допомогою iншої технiки
та бiльш простого доведення.
1. Introduction. Throughout this paper all near-rings are left near-rings. A derivation d on a near-ring
R is an additive mapping satisfying d(xy) = xd(y) + d(x)y for all x, y ∈ R. If R is a subnear-ring
of a near-ring N and d : R→ N is a map satisfies d(a+ b) = d(a)+d(b) and d(ab) = ad(b)+d(a)b
for all a, b ∈ S, where S is a nonempty subset of R, then we say that d acts as a derivation on S [1].
An element x ∈ R is called a left (right) zero divisor in R if there exists a nonzero element y ∈ R
such that xy = 0 (yx = 0). A zero divisor is either a left or a right zero divisor. By an integral
near-ring we mean a near-ring without nonzero zero divisors. A near-ring R is called a constant
near-ring, if xy = y for all x, y ∈ R and is called a zero-symmetric near-ring, if 0x = 0 for all
x ∈ R. A trivial zero-symmetric near-ring R is a zero-symmetric near-ring such that xy = y for all
x ∈ R − {0}, y ∈ R [6]. For any group (G,+), M(G) denotes the near-ring of all maps from G to
G with the two operations of addition and composition of maps. Mo(G) = {f ∈ M(G) : 0f = 0}
is the zero-symmetric subnear-ring of M(G) consists of all zero preserving maps from G to itself.
We refer the reader to the books of Meldrum [6] and Pilz [7] for basic results of near-ring theory
and their applications. We say that a near-ring R is 3-prime if, for all x, y ∈ R (xRy = {0} implies
x = 0 or y = 0). Notice that every trivial zero-symmetric near-ring is 3-prime.
In Section 2 we extend the concept of “3-prime” for subsets of a near-ring and use it to show
the nonexistence of nonzero derivation on special kinds of zero-symmetric 3-prime subnear-rings of
Mo(G). This result generalizes Theorem 1.1 due to Fong, Ke and Wang in [3].
It is easy to show that each member of the following classes has no nonzero derivations:
1. The class of all trivial zero-symmetric near-rings.
2. The class {R : R is a zero-symmetric 3-prime near-ring such that (R,+) is a cyclic group}.
3. The class {R : R is a direct sum of Ri and i ∈ Λ such that Ri is a zero-symmetric 3-prime
near-ring and (Ri,+) is a cyclic group for all i ∈ Λ}.
Let R = I × I × . . . × I = In, where I is a prime ring and n is an integer greater than two.
Define the addition on R by
c© AHMED A. M. KAMAL, KHALID H. AL-SHAALAN, 2014
420 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
NON-EXISTENCE OF NONZERO DERIVATIONS ON SOME CLASSES . . . 421
(a1, a2, . . . , an) + (b1, b2, . . . , bn) = (a1 + b1, a2 + b2, . . . , an + bn)
and define the multiplication on R by
(a1, a2, . . . , an)(b1, b2, . . . , bn) = (a1bn + b1, . . . , an−1bn + bn−1, anbn)
if (a1, a2, . . . , an) 6= (0, 0, . . . , 0) = 0 and 0(b1, b2, . . . , bn) = 0. By the same way as in Exam-
ple 2.14 of [5], this gives us a large class of zero-symmetric 3-prime near-rings which are not rings
and the zero map is the only derivation on any near-ring of the class.
2. Subsets satisfy the 3-prime condition. In this section we extend the concept of “3-prime”
for subsets. This extension will be useful in Theorem 2.1 to prove that each member of a certain
class of subnear-rings of Mo(G) has no nonzero derivations.
Definition 2.1. Let U be a nonempty subset of a near-ring R. We say that U satisfies the 3-prime
condition if, for all x, y ∈ R (xUy = {0} implies x = 0 or y = 0). We say that the element r ∈ R
satisfies the 3-prime condition if {r} satisfies the 3-prime condition.
In the next two examples we give some near-rings contain subsets satisfy the 3-prime condition.
Example 2.1. (i) Any 3-prime near-ring satisfies the 3-prime condition.
(ii) Any nonzero subset of R, where R is an integral near-ring, satisfies the 3-prime condition.
(iii) In any constant near-ring R, every element (even 0) satisfies the 3-prime condition, since
xzy = y for all x, y, z ∈ R.
Example 2.2. Let G be any group. Then M(G) and Mo(G) are near-rings have subsets satisfy
the 3-prime condition. To show that take R to be one of M(G) and Mo(G). For all g ∈ G define
βg : G → G by 0βg = 0 and tβg = g for all t ∈ G − {0}. Let B be the set {βg|g ∈ G}. Now,
suppose that fBh = {0} for some f, h ∈ R. If f 6= 0, then there exists t ∈ G such that tf 6= 0.
Therefore, tfβg = g and hence 0 = tfβgh = gh for all g ∈ G. Thus, h = 0. So B satisfies the
3-prime condition. A similar proof can be done for B1 = {βg|g ∈ G − {0}} as a subset of Mo(G)
and for the subset of all constant maps A = {αg|g ∈ G} as a subset of M(G), where tαg = g for
all t ∈ G.
Lemma 2.1. (i) Let R be a near-ring with a subset U satisfies the 3-prime condition. Then R is
3-prime. In particular, if R has an element which satisfies the 3-prime condition, then R is 3-prime.
(ii) Every subnear-ring of Mo(G) contains the subset B1 is 3-prime and every subnear-ring of
M(G) contains either B or A is 3-prime. In particular, Mo(G) and M(G) are 3-prime near-rings.
Proof. (i) If xRy = {0} for some x, y ∈ R, then xUy = {0}. Thus, either x = 0 or y = 0.
(ii) The proof is direct from Example 2.2 and (i).
If R has an element which satisfies the 3-prime condition, then R is 3-prime by Lemma 2.1(i),
but the converse need not be true as the following example shows.
Example 2.3. Let R = Mn(F ) for a field F . Then it is well-known that R is a prime ring
and for every singular matrix A of R there exists a singular nonzero matrix B such that AB = 0.
Therefore, the elements of R do not satisfy the 3-prime condition.
The following lemma extends known results about derivations on near-rings to subsets of near-
rings satisfy the 3-prime condition.
Lemma 2.2. Let R be a subnear-ring of a near-ring N with a nonzero subsemigroup U of (R, ·)
and d an additive map from R to N which acts as a derivation on U . Then
(i) For all u, v, w ∈ U, we have (ud(v) + d(u)v)w = ud(v)w + d(u)vw.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
422 AHMED A. M. KAMAL, KHALID H. AL-SHAALAN
(ii) If U satisfies the 3-prime condition on N and d(U)w = {0} for some w ∈ U, then either
d(U) = {0} or w = 0. Moreover, if R is zero-symmetric and xd(U) = {0} for some x ∈ R, then
either d(U) = {0} or x = 0.
(iii) Suppose d is a derivation on R and U satisfies the 3-prime condition on N . If d(U)x = {0}
for some x ∈ R, then either d(U) = {0} or x = 0.
Proof. (i) By the same way of the proof of Lemma 1 in [2].
(ii) Suppose d(U)w = {0}. Using (i), we have 0 = d(uv)w = ud(v)w + d(u)vw = d(u)vw for
all u, v ∈ U . Since U satisfying the 3-prime condition, we get d(U) = {0} or w = 0. The proof of
the second case is similar using that 0r = 0 for all r ∈ R.
(iii) The proof is similar to the proof of (ii) using that U = R in (i).
Remark 2.1. Let G be any group. For all g ∈ G, take βg : G → G as defined in Example 2.2.
For all g, h ∈ G, observe that βg + βh = βg+h and for all 0 6= g ∈ G, h ∈ G, we have βgβh = βh,
β0βg = 0 in Mo(G) and βhf = βhf for all f ∈ Mo(G). Let B1 as defined in Example 2.2 with
G 6= {0}. It is easy to see that B1 ∪ {0} is even a subnear-ring of the near-ring Mo(G) which is
isomorphic to the trivial zero-symmetric near-ring on G.
In Theorem 1.1 of [3], Fong, Ke and Wang had proved that any subnear-ring of Mo(G) containing
all the transformations (maps) with finite range has no nonzero derivations using the maps δx,y : G→
→ G defined by (z)δx,y = x if z = y and 0 otherwise for all x ∈ G and y ∈ G∗, whereG∗ = G−{0}.
The following theorem generalizes Theorem 1.1 of [3] with another technique and simple proof
different from the proof of it.
Theorem 2.1. Let G be any group and R a subnear-ring of Mo(G) containing B1. Suppose S
is a subset of R containing B1. If d is a map from R to Mo(G) which acts as a derivation on S and
d(0) = 0, then d(S) = {0}.
Proof. If G = {0}, then d = 0 and B1 is the empty set. So suppose that G 6= {0}. Assume that
for some 0 6= g ∈ G, d(βg) = f . If gf = h ∈ G− {0}, then
f = d(βg) = d(βgβg) = βgd(βg) + d(βg)βg =
= βgf + fβg = βgf + fβg = βh + fβg
and hence f = βh +fβg. Thus, h = gf = g(βh +fβg) = gβh +gfβg = h+g which implies g = 0,
a contradiction. Using that 0d(β0) = 0d(0) = 0, we have
gd(βg) = 0 for all g ∈ G. (2.1)
Clearly from (2.1) that βgd(βg) = 0 for all g ∈ G. Thus, d(βg) = d(βgβg) = βgd(βg) + d(βg)βg =
= d(βg)βg for all g ∈ G. It follows that Gd(βg) = Gd(βg)βg and hence
Gd(βg) ⊆ {0, g} for all g ∈ G. (2.2)
If d(βg) = 0 for some g ∈ G− {0}, then we claim first that d(B1) = {0} in Mo(G). Indeed, for all
h ∈ G− {0}, we get
0 = d(βg) = d(βhβg) = βhd(βg) + d(βh)βg = d(βh)βg.
Thus, d(B1)βg = {0}. But B1 is a subsemigroup of Mo(G) satisfying the 3-prime condition and
βg is a nonzero element. Therefore, d(B1) = {0} by using Lemma 2.2(ii). After that, we claim that
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
NON-EXISTENCE OF NONZERO DERIVATIONS ON SOME CLASSES . . . 423
d(S) = {0}. Indeed, for all s ∈ S, g ∈ G− {0}, we obtain d(βgs) = 0 (even for gs = 0). It follows
that
0 = d(βgs) = d(βgs) = βgd(s) + d(βg)s = βgd(s) = βhβgd(s)
for some h ∈ G− {0}. Since B1 satisfies the 3-prime condition, we have d(s) = 0 for all s ∈ S and
d(S) = {0}.
To complete the proof we will show that d(βg) 6= 0 for all g ∈ G − {0} is impossible. If
G = {0, g}, then d(βg) = 0 since gd(βg) = 0 by (2.1).
Now, suppose G contains more than two elements and d(βg) 6= 0 for all g ∈ G − {0}. Thus,
from (2.1) and (2.2), we obtain
for all g ∈ G− {0} there exists h ∈ G− {0, g} such that hd(βg) = g. (2.3)
Observe that
d(βg) = d(βhβg) = βhd(βg) + d(βh)βg. (2.4)
Using gd(βg) = 0 and (2.4), we have for all g ∈ G− {0}
0 = g(βhd(βg) + d(βh)βg) = hd(βg) + gd(βh)βg = g + gd(βh)βg. (2.5)
Using (2.2), we have Gd(βh) ⊆ {0, h}. If gd(βh) = 0, then g = 0 from (2.5), a contradiction. It
follows that gd(βh) = h. Hence, (2.5) gives us that g + g = 0 for all g ∈ G and so G is a 2-torsion
group. From (2.4), we have
(g + h)d(βg) = (g + h)βhd(βg) + (g + h)d(βh)βg. (2.6)
If g+h = 0, then g = −h = h which is a contradiction with (2.3). Thus, we have (g+h)βh = h.
From (2.3), equation (2.6) will be
(g + h)d(βg) = hd(βg) + (g + h)d(βh)βg = g + (g + h)d(βh)βg. (2.7)
Using (2.2) and (2.7), if (g + h)d(βg) = 0, then g + (g + h)d(βh)βg = 0 which means (g +
+ h)d(βh)βg = −g = g. Thus, (g + h)d(βh) = h. In the other case, if (g + h)d(βg) = g, then
(g + h)d(βh)βg = 0 and hence (g + h)d(βh) = 0. Therefore, (2.7) implies that (g + h)d(βg) + (g +
+ h)d(βh) equal either g or h. On the other hand, from (2.1), we have
0 = (g + h)d(βg+h) = (g + h)d(βg + βh) = (g + h)[d(βg) + d(βh)] =
= (g + h)d(βg) + (g + h)d(βh).
Thus, g = 0 or h = 0 which is a contradiction with g 6= 0 and h 6= 0. Therefore, d(βg) 6= 0 for all
g ∈ G− {0} is impossible.
Theorem 2.1 is proved.
Observe that B1 is a proper subset of the set of all transformations with finite range of Mo(G).
In particular, if G is finite, then
∑
x∈G∗ δg,x = βg. Therefore, Theorem 2.1 generalizes Theorem 1.1
of [3] (in the sense that the class of zero-symmetric 3-prime subnear-rings of Mo(G) in Theorem 2.1
is larger than the class of subnear-rings of Mo(G) in Theorem 1.1 of [3]).
Corollary 2.1. Let G be any group. Any subnear-ring of Mo(G) containing B1 has no nonzero
derivation. In particular, Mo(G) has no nonzero derivation.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
424 AHMED A. M. KAMAL, KHALID H. AL-SHAALAN
The following example shows that the condition ”the subnear-ring of Mo(G) containing the
subset B1” in Theorem 2.1 and Corollary 2.1 is not redundant.
Example 2.4. Take the near-ring R = {f ∈ Mo(Z4) : {2, 3}f = {0}} = AnnMo(Z4) ({2, 3}) as
a special case of Example 2.7 in [5]. Then R is a subnear-ring of Mo(Z4) which is not a ring. Define
D : R → Mo(Z4) by D(fy) = f2y for all y ∈ Z4. By the same way as in Example 2.7 of [5], we
obtain that D acts as a nonzero derivation on R. Notice that B1 * R.
Remark 2.2. Since for any group G, we have any subnear-ring R of Mo(G) containing the
subset B1 is a 3-prime near-ring by Lemma 2.1(ii) and has no nonzero derivation by Corollary 2.1.
Therefore, we have a very large class of zero-symmetric 3-prime near-rings which are not rings such
that every near-ring of the class has no nonzero derivation.
1. Bell H. E., Argac N. Derivations, products of derivations and commutativity in near-rings // Algebra Colloq. –
2001. – 8. – P. 399 – 407.
2. Bell H. E., Mason G. On derivations in near-rings // Proc. Near-rings and Near-fields / Eds G. Betsch et al. North-
Holland Math. Stud. – 1987. – P. 31 – 35.
3. Fong Y., Ke W.-F., Wang C.-S. Nonexistence of derivations on transformation near-rings // Communs Algebra. – 2000. –
28. – P. 1423 – 1428.
4. Hungerford T. W. Algebra. – New York: Holt, Rinehart and Winston, Inc., 1974.
5. Kamal Ahmed A. M., Al-Shaalan Khalid H. Existence of derivations on near-rings // Math. Slovaca. – 2013. – 63,
№ 3. – P. 431 – 448.
6. Meldrum J. D. P. Near-rings and their links with groups. – Boston, MA: Pitman, 1985.
7. Pilz G. Near-rings. – Amsterdam: North-Holland, 1983.
Received 06.04.12,
after revision — 19.11.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
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| spelling | umjimathkievua-article-21442019-12-05T10:24:58Z Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings Неіснування ненульових похідних на деяких класах 3-простих майже-кілець з нульовою симетрію Al-Shaalan, Khalid H. Kamal, Ahmed A. M. Аль-Шаалан, Халід Г. Камаль, Ахмед А. М. We give some classes of zero-symmetric 3-prime near-rings such that every member of these classes has no nonzero derivation. Moreover, we extend the concept of “3-prime” to subsets of near-rings and use it to generalize Theorem 1.1 due to Fong, Ke, and Wang concerning the transformation near-rings M o (G) by using a different technique and a simpler proof. Наведено дєякі класи 3-простих майже-кілець з нульовою симєтрією таких, що будь-який елемент цих класів не має ненульової похідної. Крім того, поняття „3-простих" узагальнено на підмножини майже-кілець i застосовано, щоб узагальнити теорему 1.1 Фонга, Ке i Ванга про трансформацію майже-кілець M o (G) за допомогою іншої техніки та більш простого доведення. Institute of Mathematics, NAS of Ukraine 2014-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2144 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 3 (2014); 420–424 Український математичний журнал; Том 66 № 3 (2014); 420–424 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2144/1290 https://umj.imath.kiev.ua/index.php/umj/article/view/2144/1291 Copyright (c) 2014 Al-Shaalan Khalid H.; Kamal Ahmed A. M. |
| spellingShingle | Al-Shaalan, Khalid H. Kamal, Ahmed A. M. Аль-Шаалан, Халід Г. Камаль, Ахмед А. М. Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings |
| title | Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings |
| title_alt | Неіснування ненульових похідних на деяких класах 3-простих майже-кілець з нульовою симетрію |
| title_full | Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings |
| title_fullStr | Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings |
| title_full_unstemmed | Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings |
| title_short | Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings |
| title_sort | nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2144 |
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