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

Ahmed A. M. Kamal

Khalid H. Al-Shaalan

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 512.5Ahmed 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 CLASSESOF ZERO-SYMMETRIC 3-PRIME NEAR-RINGSНЕIСНУВАННЯ НЕНУЛЬОВИХ ПОХIДНИХ НА ДЕЯКИХ КЛАСАХ3-ПРОСТИХ МАЙЖЕ-КIЛЕЦЬ З НУЛЬОВОЮ СИМЕТРIЄЮWe give some classes of zero-symmetric 3-prime near-rings such that every member in these classes has no nonzeroderivation. Moreover, we extend the concept of ”3-prime” to subsets of near-rings and use it to generalize Theorem 1.1 dueto Fong, Ke, and Wang concerning the transformation near-rings Mo(G) by using a different technique and a more simpleproof.Наведено деякi класи 3-простих майже-кiлець з нульовою симетрiєю таких, що будь-який елемент цих класiв не маєненульової похiдної. Крiм того, поняття „3-простих“ узагальнено на пiдмножини майже-кiлець i застосовано, щобузагальнити теорему 1.1 Фонга, Ке i Ванга про трансформацiю майже-кiлець Mo(G) за допомогою iншої технiкита бiльш простого доведення.1. Introduction. Throughout this paper all near-rings are left near-rings. A derivation d on a near-ringR is an additive mapping satisfying d(xy) = xd(y) + d(x)y for all x, y ∈ R. If R is a subnear-ringof 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)bfor 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 ∈ Rsuch that xy = 0 (yx = 0). A zero divisor is either a left or a right zero divisor. By an integralnear-ring we mean a near-ring without nonzero zero divisors. A near-ring R is called a constantnear-ring, if xy = y for all x, y ∈ R and is called a zero-symmetric near-ring, if 0x = 0 for allx ∈ R. A trivial zero-symmetric near-ring R is a zero-symmetric near-ring such that xy = y for allx ∈ R − {0}, y ∈ R [6]. For any group (G,+), M(G) denotes the near-ring of all maps from G toG 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 theoryand their applications. We say that a near-ring R is 3-prime if, for all x, y ∈ R (xRy = {0} impliesx = 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 showthe nonexistence of nonzero derivation on special kinds of zero-symmetric 3-prime subnear-rings ofMo(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-primenear-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 byc© AHMED A. M. KAMAL, KHALID H. AL-SHAALAN, 2014420 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3NON-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 ringsand 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 certainclass 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-primecondition if, for all x, y ∈ R (xUy = {0} implies x = 0 or y = 0). We say that the element r ∈ Rsatisfies 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, sincexzy = 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 satisfythe 3-prime condition. To show that take R to be one of M(G) and Mo(G). For all g ∈ G defineβg : 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 the3-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 forall t ∈ G.Lemma 2.1. (i) Let R be a near-ring with a subset U satisfies the 3-prime condition. Then R is3-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 ofM(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 ringand 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, № 3422 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 eitherd(U) = {0} or w = 0. Moreover, if R is zero-symmetric and xd(U) = {0} for some x ∈ R, theneither 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 forall u, v ∈ U . Since U satisfying the 3-prime condition, we get d(U) = {0} or w = 0. The proof ofthe 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 withG 6= {0}. It is easy to see that B1 ∪ {0} is even a subnear-ring of the near-ring Mo(G) which isisomorphic 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) containingall 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 proofdifferent from the proof of it.Theorem 2.1. Let G be any group and R a subnear-ring of Mo(G) containing B1. Suppose Sis a subset of R containing B1. If d is a map from R to Mo(G) which acts as a derivation on S andd(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 thatfor some 0 6= g ∈ G, d(βg) = f . If gf = h ∈ G− {0}, thenf = d(βg) = d(βgβg) = βgd(βg) + d(βg)βg == βgf + fβg = βgf + fβg = βh + fβgand 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 havegd(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 henceGd(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 allh ∈ G− {0}, we get0 = 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 andβg is a nonzero element. Therefore, d(B1) = {0} by using Lemma 2.2(ii). After that, we claim thatISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3NON-EXISTENCE OF NONZERO DERIVATIONS ON SOME CLASSES . . . 423d(S) = {0}. Indeed, for all s ∈ S, g ∈ G− {0}, we obtain d(βgs) = 0 (even for gs = 0). It followsthat0 = 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 andd(S) = {0}.To complete the proof we will show that d(βg) 6= 0 for all g ∈ G − {0} is impossible. IfG = {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 obtainfor all g ∈ G− {0} there exists h ∈ G− {0, g} such that hd(βg) = g. (2.3)Observe thatd(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. Itfollows that gd(βh) = h. Hence, (2.5) gives us that g + g = 0 for all g ∈ G and so G is a 2-torsiongroup. 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 have0 = (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 allg ∈ 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 1.1of [3] (in the sense that the class of zero-symmetric 3-prime subnear-rings of Mo(G) in Theorem 2.1is 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 nonzeroderivation. In particular, Mo(G) has no nonzero derivation.ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3424 AHMED A. M. KAMAL, KHALID H. AL-SHAALANThe following example shows that the condition ”the subnear-ring of Mo(G) containing thesubset 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}) asa special case of Example 2.7 in [5]. Then R is a subnear-ring of Mo(Z4) which is not a ring. DefineD : R → Mo(Z4) by D(fy) = f2y for all y ∈ Z4. By the same way as in Example 2.7 of [5], weobtain 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 thesubset 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 suchthat 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 — 19.11.13ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3

Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings

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Nonexistence of nonzero derivations on some classes of zero-symmetric 3-prime near-rings

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)