Semigroup ideals with semiderivations in 3-prime near-rings

The purpose of this paper is to obtain the structure of certain near-rings satisfying the following conditions: (i) d(I) ⊆ Z(N), (ii) d(−I) ⊆ Z(N), (iii) d([x, y]) = 0, (iv) d([x, y]) = [x, y], (v) d(x ◦ y) = 0, (vi) d(x ◦ y) = x ◦ y for all x, y ∈ I , with I is a semigroup ideal and d is a semiderivation associated with an automorphism. Furthermore; an example is given to illustrate that the 3-primeness hypothesis is not superfluous. Definitions and terminology In this paper N will denote a zero symmetric left near-ring. For any x, y ∈ N the symbol [x, y] will denote the commutator xy − yx, while the symbol x ◦ y will stand for the anti-commutator xy + yx. The symbol Z(N) will represent the multiplicative center of N , that is, Z(N) = {x ∈ N | xy = yx for all y ∈ N}. Unless specified, we will use the word near-ring to mean zero symmetric left near-ring. A near-ring N is said to be 3-prime if xNy = {0} for all x, y ∈ N implies x = 0 or y = 0. A nonempty subset I of N is called a semigroup right ideal (resp. semigroup left ideal) if IN ⊂ I (resp. NI ⊂ I); and if I is both a semigroup right and a semigroup left ideal, then I is said to be a semigroup ideal. N is said to be 2-torsion free if x ∈ N and 2x = 0 implies x = 0. An additive mapping δ : N → N is called a derivation if δ(xy) = δ(x)y + xδ(y) holds for all x, y ∈ N . Let g be an additive mapping of N , an additive mapping d : N → N is called a semiderivation of N associated with g if d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and d(g(x)) = g(d(x)) for all x, y ∈ N , or equivalently, as noted in [10], 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 . In the case of rings, semiderivations have received significant attention in recent years. We prove that some theorems in the setting of a semigroup ideal of a 3-prime near ring admitting a semiderivation, and thereby extend some known results [4, Theorem 2.1], [8, Theorem 2.6] and [8, Theorem 2.9]. 1 Main Results In this paper, the semiderivations used are associated with automorphisms. To prove our main theorems, we need the following lemmas. Lemma 1. [4, Lemma 1.4(i)] Let N be a 3-prime near-ring, and I a nonzero semigroup ideal of N . If x, y ∈ N and xIy = {0}, then x = 0 or y = 0. Lemma 2. 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) [2, Lemma 1.5] If N ⊆ Z(N), then N is a commutative ring. Lemma 3. Let N be a near-ring and d is a semiderivation of N . Then N satisfies the following partial distributive law i) ( d(x)y + g(x)d(y) ) z = d(x)yz + g(x)d(y)z for all x, y, z ∈ N. ii) ( xd(y) + d(x)g(y) ) z = xd(y)z + d(x)g(y)z for all x, y, z ∈ N. Semigroup ideals with semiderivations in 3-prime near-rings 439 Theorem 1. Let N be a 3-prime near-ring and I be a nonzero semigroup ideal of N . If N admits a nonzero semiderivation d, then the following assertions are equivalent i) d(I) ⊆ Z(N) ii) N is a commutative ring. Proof. ii)⇒ i) is obvious. i)⇒ ii) by the hypothesis given, we have d(xy)z = zd(xy) for all y ∈ I, x, z ∈ N. Taking Lemma 3(i), we get d(x)yz + g(x)d(y)z = zd(x)y + zg(x)d(y) for all y ∈ I, x, z ∈ N. So that, d(x)yz + d(y)g(x)z = zd(x)y + d(y)zg(x) for all y ∈ I, x, z ∈ N. (1.1) Substituting g(x) for z in (1.1), we obtain d(x)yg(x) = g(x)d(x)y for all y ∈ I, x ∈ N. (1.2) Replacing y by yt in (1.2) and using this, we get d(x)ytg(x) = (g(x)d(x)y)t = d(x)yg(x)t for all y ∈ I, x, t ∈ N. The last equation shows that d(x)y[g(x), t] = 0 for all y ∈ I, x, t ∈ N this means that d(x)I[g(x), t] = {0} for all x, t ∈ N. By Lemma 1, this implies that d(x) = 0 or g(x) ∈ Z(N) for all x ∈ N. (1.3) Taking the fact that d 6= 0, then (1.3) shows that there is an element x0 ∈ N such as g(x0) ∈ Z(N) and d(x0) 6= 0. In this case, equation (1.1) yields d(x0)yz = zd(x0)y for all y ∈ I, z ∈ N. Again replacing y by yt, we get d(x0)ytz = (zd(x0)y)t = d(x0)yzt for all y ∈ I, z, t ∈ N. Hence, d(x0)y[z, t] = 0 for all y ∈ I, z, t ∈ N this is reduced to d(x0)I[z, t] = {0} for all z, t ∈ N. (1.4) Taking Lemma 1, (1.4) implies that d(x0) = 0 or N ⊂ Z(N). And since the first of these conditions is impossible, the second must hold N a commutative ring by Lemma 2(ii). Corollary 1. [4, Theorem 2.1] Let N be a 3-prime near-ring, and let I be a nonzero semigroup ideal of N . If N admits a nonzero derivation d for which d(I) ⊆ Z(N), then N is a commutative ring. Theorem 2. Let N be a 2-torsion free 3-prime near-ring and I be a nonzero semigroup ideal of N . If N admits a nonzero semiderivation d, then the following assertions are equivalent i) d(−I) ⊆ Z(N) ii) N is a commutative ring. 440 A. Boua and L. Oukhtite and A. Raji Proof. For ii)⇒ i), the proof is obvious. i)⇒ ii), we have d(−x) ∈ Z(N) for all x ∈ I , then d(−tx) = d(t(−x)) ∈ Z(N) for all x ∈ I, t ∈ N. (1.5) In particular, for all t ∈ Z(N) we have d(t(−x)) = td(−x) + d(t)g(−x) ∈ Z(N) for all x ∈ I. by Lemma 3(ii), we obtain d(t)g(−x) ∈ Z(N) for all x ∈ I. (1.6) Since g is an automorphism, then d(t) ∈ Z(N). By the application of Lemma 2(i), (1.6) yields d(t) = 0 or g(−x) ∈ Z(N) for all x ∈ I, t ∈ Z(N). (1.7) If d(Z(N)) = {0}, taking (1.5) into account, we get d ( d ( t(−x) )) = 0 for all x ∈ I, t ∈ N. So that, d2(t)(−x) + 2g(d(t))d(−x) = 0 for all x ∈ I, t ∈ N. (1.8) Replacing t by d(t) in (1.8), we get d3(t)(−x) + 2g(d2(t))d(−x) = 0 for all x ∈ I, t ∈ N. (1.9) on the other hand, applying d for (1.8), we find that d3(t)(−x) + 3g(d2(t))d(−x) = 0 for all x ∈ I, t ∈ N. (1.10) From (1.9) and (1.10), we conclude that g(d2(t))d(−x) = 0 for all x ∈ I, t ∈ N . Taking the fact that d(−x) ∈ Z(N), then d2 ( g(t) ) Nd(−x) = {0} for all x ∈ I, t ∈ N. In the light of the 3-primeness of N , the last equation implies that d2 = 0 or d = 0. (1.11) If d2 = 0, then d = 0 (see prove of Theorem 2 in [10]), and therefore (1.11) shows that d = 0, a contradiction. Consequently d(Z(N)) 6= {0} and (1.7) prove that g(−x) ∈ Z(N) for all x ∈ I. Let v ∈ N and x ∈ I , we have g(−vx) = g(v)g(−x) ∈ Z(N), by Lemma 2(i), we get g(−x) = 0 or g(v) ∈ Z(N) for all x ∈ I, v ∈ N. (1.12) i) If g(−x) = 0 for all x ∈ I , by this hypothesis we have d(−yx) = d(y)g(−x) + yd(−x) ∈ Z(N) for all x ∈ I, y ∈ N. So that, yd(−x) ∈ Z(N) for all x ∈ I, y ∈ N. Using Lemma 2(i) and taking the fact d 6= 0, we arrive at N ⊂ Z(N). Applying Lemma 2(ii), we conclude that N is a commutative ring. ii) If there is an element x0 ∈ I such that g(−x0) 6= 0, then equation (1.12) shows that g(v) ∈ Z(N) for all v ∈ N . Since g is an automorphism we conclude that N ⊂ Z(N). Thus N is a commutative ring. This completes the proof of our theorem. Corollary 2. [8, Lemma 2.4] Let N be a 2-torsion free 3-prime near-ring and I be a nonzero semigroup ideal of N . If N admits a nonzero derivation d for which d(−I) ⊆ Z(N), then N is a commutative ring. Theorem 3. Let N be a 3-prime near-ring and I be a nonzero semigroup ideal of N . If N admits a semiderivation d, then the following assertions are equivalent: i) d([x, y]) = 0 for all x, y ∈ I . Semigroup ideals with semiderivations in 3-prime near-rings 441 ii) d([x, y]) = [x, y] for all x, y ∈ I . iii) N is a commutative ring. Proof. iii)⇒ i) and iii)⇒ ii) are obvious. Proving that i)⇒ iii). Suppose that d([x, y]) = 0 for all x, y ∈ I. (1.13) Substituting xy for y in (1.13), we have d(x)[x, y] + g(x)d[x, y] = 0 for all x, y ∈ I. Hence, d(x)xy = d(x)yx for all x, y ∈ I. (1.14) Replacing y by yt in (1.14) and using this, we get d(x)I[x, t] = {0} for all x ∈ I, t ∈ N. Taking into account the Lemma 1, we get d(x) = 0 or x ∈ Z(N) for all x ∈ I. (1.15) Since d is associated with an automorphism, we have d(x) ∈ Z(N) for each x ∈ Z(N), then (1.15) illustrated d(I) ⊆ Z(N). By the use of Theorem 1, we obtain N is a commutative ring. Proving that ii)⇒ iii). By the hypothesis given, we have d([x, y]) = [x, y] for all x, y ∈ I. (1.16) Replacing y by xy in (1.16), we get xd([x, y]) + d(x)g([x, y]) = x[x, y] for all x, y ∈ I. It follows that d(x)g(x)g(y) = d(x)g(y)g(x) for all x, y ∈ I. (1.17) Since g is an automorphism,(1.17) shows that d(x)g(x)j = d(x)jg(x) for all x ∈ I, j ∈ J (1.18) with J = g(I), it is clear that J is a semigroup ideal of N. Substituting jz for j in (1.18) and using this, we obtain d(x)j[g(x), z] = 0 for all x ∈ I, j ∈ J, z ∈ N. (1.19) Thus, d(x)J [g(x), z] = {0} for all x ∈ I, z ∈ N. (1.20) By the application of Lemma 1, (1.20) yields that d(x) = 0 or g(x) ∈ Z(N) for all x ∈ I. Which implies that d(g(x)) ∈ Z(N) for all x ∈ I Consequently, we deduce that d(J) ⊂ Z(N). And therefore, Theorem 1 assures that N is a commutative ring. This completes the proof of our theorem. Corollary 3. [4, Theorem 4.1] Let N be a 3-prime near-ring, and U a nonzero semigroup ideal. If N admits a derivation d such that d2 6= 0 and d(uv) = d(vu) for all u, v ∈ U , then N is a commutative ring. Corollary 4. Let N be a 3-prime near-ring. If N admits a nonzero derivation d such that d([x, y]) = 0 for all x, y ∈ N , then N is a commutative ring. Corollary 5. [8, Theorem 2.6] Let N be a 3-prime near-ring and I be a nonzero semigroup ideal of N . If N admits a nonzero derivation d such that d([x, y]) = [x, y] for all x, y ∈ I , then N is a commutative ring. 442 A. Boua and L. Oukhtite and A. Raji Corollary 6. [7, Theorem 2.2] Let N be a 3-prime near-ring. If N admits a nonzero derivation d such that d([x, y]) = [x, y] for all x, y ∈ N , then N is a commutative ring. Now, replacing the commutator [x, y] by the anti-commutator x ◦ y, our aim is to study this issue and to see if the results are different. Theorem 4. Let N be a 2-torsion free 3-prime near-ring and I be a semigroup ideal of N , then N admits no nonzero semiderivation d satisfying one of the assertions as the following: i) d(x ◦ y) = 0 for all x, y ∈ I. ii) d(x ◦ y) = x ◦ y for all x, y ∈ I. Proof. i) Suppose that there is d which indicates the following d(x ◦ y) = 0 for all x, y ∈ I. (1.21) Replacing y by xy in (1.21) and taking the fact that x ◦ xy = x(x ◦ y), we get d(x)(x ◦ y) = 0 for all x, y ∈ I. So that, d(x)xy = −d(x)yx for all x, y ∈ I. (1.22) Substituting yt for y in (1.22), we obtain d(x)y(−x)t = d(x)yt(−x) for all x, y ∈ I, t ∈ N which can be rewritten as d(x)I[−x, t] = {0} for all x ∈ I, t ∈ N. By using the lemma 1, we have d(x) = 0 or − x ∈ Z(N) for all x ∈ I. Hence, d(−x) ∈ Z(N) for all x ∈ I , it means that d(−I) ⊂ Z(N). According to theorem 2, we get N is a commutative ring. In this case, returning to the hypothesis given, we have d(xy) = 0 for all x, y ∈ I. It follows that d(x)y + g(x)d(y) = 0 for all x, y ∈ I (1.23) Taking yz instead of y in (1.23), we get d(x)yz = 0 for all x, y, z ∈ I. Therefore, d(x)Iz = {0} for all x, z ∈ I. By Lemma 1, the last expression shows that d = 0, a contradiction. ii) Suppose there is d such that d(x ◦ y) = x ◦ y for all x, y ∈ I. (1.24) Putting xy instead of y in (1.24), we arrive at d(x)g(x)g(y) = −d(x)g(y)g(x) for all x, y ∈ I. Which implies that d(x)g(x)n = −d(x)ng(x) for all x ∈ I, n ∈ J = g(I). (1.25) Writing nm instead of n in (1.25), we find that d(x)n[g(−x),m] = 0 for all x ∈ I, n ∈ J,m ∈ N implying d(x)J [g(−x),m] = {0} for all x ∈ I,m ∈ N. (1.26) Semigroup ideals with semiderivations in 3-prime near-rings 443 Applying Lemma 1, (1.26) shows that d(x) = 0 or g(−x) ∈ Z(N) for all x ∈ I. And therefore d(g(−x)) ∈ Z(N) for all x ∈ I , then d(−J) ⊆ Z(N). According to Theorem 2, we conclude that N is a commutative ring. In this case, returning to the hypothesis given, we have d(xy) = xy for all x, y ∈ I it follows that d(x)y + g(x)d(y) = xy for all x, y ∈ I. (1.27) Substituting xz for x in (1.27), we obtain g(x)g(z)d(y) = 0 for all x, y, z ∈ I. Which can be rewritten as g(x)Jd(y) = {0} for all x, y ∈ I. (1.28) By Lemma 1, (1.28) demonstrates g(I) = {0} or d = 0, but each of these conditions yields a contradiction. Corollary 7. [8, Theorem 2.9] Let N be a 2-torsion free 3-prime near-ring, and I be a nonzero semigroup ideal of N . Then there is no derivation d such that d(x ◦ y) = x ◦ y for all x, y ∈ I. The following example shows that the primeness is necessary in the hypotheses of the above theorems. Example Let S be a 2-torsion free noncommutative near-ring. Let us defineN and d, g : N → N by: N = { 0 0 x 0 0 y 0 0 0  | x, y ∈ S}. d  0 0 x 0 0 y 0 0 0  =  0 0 x 0 0 0 0 0 0  , g  0 0 x 0 0 y 0 0 0  =  0 0 y 0 0 x 0 0 0  . Then, it is straightforward to check that N is not 3-prime left near-ring admitting a nonzero semiderivation d associated with g. Moreover; it is easy to verify that d satisfies the properties: i) d(N) ⊆ Z(N) ii) d(−N) ⊆ Z(N) iii) d([A,B]) = 0 iv) d([A,B]) = [A,B] v) d(A ◦B) = 0 vi) d(A ◦B) = A ◦B for all A,B ∈ N . However, N is not commutative.