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«Commentationes Mathematicae Universitatis Carolinae Abdelkarim Boua; Lahcen Oukhtite; Abderrahmane Raji Jordan ideals and derivations in prime ...»

Commentationes Mathematicae Universitatis Carolinae

Abdelkarim Boua; Lahcen Oukhtite; Abderrahmane Raji

Jordan ideals and derivations in prime near-rings

Commentationes Mathematicae Universitatis Carolinae, Vol. 55 (2014), No. 2, 131--139

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This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Comment.Math.Univ.Carolin. 55,2 (2014) 131–139 Jordan ideals and derivations in prime near-rings Abdelkarim Boua, Lahcen Oukhtite, Abderrahmane Raji Abstract. In this paper we investigate 3-prime near-rings with derivations satisfying certain differential identities on Jordan ideals, and we provide examples to show that the assumed restrictions cannot be relaxed.

Keywords: prime near-rings; Jordan ideals; derivations; commutativity Classification: 16N60, 16W25, 16Y30

1. Introduction Throughout this paper N will be a zero-symmetric right near-ring, and usually N will be 3-prime, that is, will have the property that xN y = 0 for x, y ∈ N implies x = 0 or y = 0. The symbol Z(N ) will denote the multiplicative center of N. A near-ring N is called zero-symmetric if x0 = 0, for all x ∈ N (recall that right distributivity yields 0x = 0). An additive mapping d : N −→ N is said to be a derivation if d(xy) = xd(y) + d(x)y for all x, y ∈ N, or equivalently, as noted in [15], that d(xy) = d(x)y + xd(y) for all x, y ∈ N. We will write for all x, y ∈ N, [x, y] = xy − yx and x ◦ y = xy + yx for the Lie product and Jordan product, respectively. Recall that N is called 2-torsion free if 2x = 0 implies x = 0 for all x ∈ N.

Many results in literature indicate how the global structure of a ring is often tightly connected to the behavior of additive mappings defined on that ring.

In this direction, several authors have studied the structure of prime and semiprime rings admitting suitably constrained additive mappings, as automorphisms, derivations, skew derivations and generalized derivations acting on appropriate subsets of the rings. Moreover, many of the obtained results extend those proven previously just for the action of the considered mappings on the whole ring to their actions on appropriate subsets of the ring. Furthermore, many authors have proved analogous results for prime and semiprime near-rings (see [2], [3], [4], [7], [8], [15] etc). Recently, there has been a great deal of work concerning commutativity of prime rings with additive mappings satisfying certain differential identities involving Jordan ideals (see [11], [12], [13], [14], [16]). Here we continue this line of investigation and we study the structure of 3-prime near-rings in which derivations satisfy certain identities involving Jordan ideal. Indeed, motivated by the concepts of Jordan ideals on rings, here we initiate the concepts of Jordan

ideals on near-rings as follows:

132 A. Boua, L. Oukhtite, A. Raji Definition 1. Let N be a near-ring. An additive subgroup J of N is said to be a Jordan ideal of N if j ◦ n ∈ J and n ◦ j ∈ J for all j ∈ J, n ∈ N.

Example 1. Define two operations “+” and “.” on Z/4Z by:

–  –  –

It is easy to check that (Z/4Z, +,.) is a right 3-prime near-ring. Moreover, if we set J = {0, 1}, then J is a Jordan ideal of Z/4Z.

2. Conditions on Jordan ideals Our aim in this section is to prove that if a Jordan ideal satisfies suitable conditions, then the near-ring must be a commutative ring. We leave the proof of the following easy lemmas to the reader.

Lemma 1. Let N be a 3-prime near-ring and J a nonzero Jordan ideal of N.

If Jx = {0}, then x = 0.

Lemma 2. Let N be a 3-prime near ring and J a Jordan ideal of N.

If j 2 = 0 for all j ∈ J, then J = 0.

It is well known that a 2-torsion free 3-prime ring must be commutative if it admits a nonzero central Jordan ideal. The following lemma gives an analogous result for near-rings.

Lemma 3. Let N be a 2-torsion free 3-prime near-ring and J a nonzero Jordan ideal of N.

If J ⊆ Z(N ), then N is a commutative ring.

Proof: From

–  –  –

proving that N is a commutative ring.

Remark 1. In ring theory it is known that a 2-torsion free 3-prime ring must be commutative if it admits a nonzero commutative Jordan ideal.

In the case of a 2-torsion free 3-prime near-ring N, the assumption that [J, J] = 0 yields (i + j)(k + k) = (k + k)(i + j) so that

k((j + i) − (i + j)) = 0 for all i, j, k ∈ J

and application of Lemma 1 yields j + i = i + j for all i, j ∈ J. It seems that the near-ring need not be a commutative ring but we are unable to construct a counter-example. Hence, we leave it as an open question.

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.

Proof: (i) Assume that

–  –  –

Once again using the 3-primeness of N, equation (4) implies that either k ∈ Z(N ) or k = 0 for all k ∈ J and therefore J ⊆ Z(N ). According to Lemma 3, we conclude that N is a commutative ring.

(ii) Suppose that

–  –  –

Since equation (7) is the same as equation (3), then arguing as in the first case we get the required result.

3. Conditions with derivations Theorem 2. Let N be a 2-torsion free 3-prime near-ring and J a nonzero Jordan ideal of N. If N admits a nonzero derivation d such that [d(n), j] = 0 for all n ∈ N, j ∈ J, then N is a commutative ring.

Proof: We are given that

–  –  –

but N is not a commutative ring.

Theorem 3. Let N be a 2-torsion free 3-prime near-ring and J a nonzero Jordan ideal of N.

If N admits a nonzero derivation d such that d([n, j]) = 0 for all n ∈ N, j ∈ J, then N is a commutative ring.

Proof: Assume that

–  –  –

If there exists j0 ∈ J such that d(j0 ) = 0, then from d(j0 ◦ j0 ) = j0 ◦ j0 it follows that j0 = 0. Since d(n ◦ j0 ) = n ◦ j0, then replacing n by nj0 in this equation we get

–  –  –

As N is 3-prime, we conclude that j0 = 0. Accordingly, equation (19) reduces to J ⊆ Z(N ) and Lemma 3 assures that N is a commutative ring. Hence, by 2-torsion freeness, equation (17) becomes

–  –  –

Using similar arguments as used previously we arrive at J ⊆ Z(N ) and application of Lemma 3 implies that N is a commutative ring. Hence equation (22) together with 2-torsion freeness forces

–  –  –

which leads to d = 0; a contradiction.

(ii) Using similar techniques, we get the required result.

Remark 2. The results in this paper remain true for left near-rings with the obvious changes.

References

[1] Argac N., On prime and semiprime rings with derivations, Algebra Colloq. 13 (2006), no. 3, 371–380.

[2] Ashraf M., Ali A., Ali S., (σ, τ )-derivations on prime near rings, Arch. Math. (Brno) 40 (2004), no. 3, 281–286.

[3] Ashraf M., Shakir A., On (σ, τ )-derivations of prime near-rings II, Sarajevo J. Math. 4 (2008), no. 16, 23–30.

[4] Beidar K.I., Fong Y., Wang X.K., Posner and Herstein theorems for derivations of 3-prime near-rings, Comm. Algebra 24 (1996), no. 5, 1581–1589.

[5] Bell H.E., On derivations in near-rings, II, Kluwer Academic Publishers, Dordrecht, 1997, pp. 191–197.

[6] Bell H.E., Argac N., Derivations, products of derivations, and commutativity in near-rings, Algebra Colloq. 8 (2001), no. 4, 399–407.

Jordan ideals and derivations in prime near-rings [7] Bell H.E., Mason G., On derivations in near-rings, North-Holland Math. Stud., 137, NorthHolland, Amsterdam, 1987, pp. 31–35.

[8] Bell H.E., Mason G., On derivations in near-rings and rings, Math. J. Okayama Univ. 34 (1992), 135–144.

[9] Boua A., Oukhtite L., Derivations on prime near-rings, Int. J. Open Probl. Comput. Sci.

Math. 4 (2011), no. 2, 162–167.

[10] Mansoor A., Lie and Jordan ideals in prime rings with derivations, Proc. Amer. Math.

Soc. 55 (1976), no. 2, 275–278.

[11] Oukhtite L., Mamouni A., Derivations satisfying certain algebraic identities on Jordan ideals, Arab. J. Math. 1 (2012), no. 3, 341–346.

[12] Oukhtite L., Posner’s second theorem for Jordan ideals in rings with involution, Expo.

Math. 29 (2011), no. 4, 415–419.

[13] Oukhtite L., Left multipliers and Jordan ideals in rings with involution, Afr. Diaspora J.

Math. 11 (2011), no. 1, 24–28.

[14] Oukhtite L., On Jordan ideals and derivations in rings with involution, Comment. Math.

Univ. Carolin. 51 (2010), no. 3, 389–395.

[15] Wang X.K., Derivations in prime near-rings, Proc. Amer. Math. Soc. 121 (1994), 361–366.

[16] Zaidi S.M.A., Ashraf M., Ali S., On Jordan ideals and left (θ, θ)-derivations in prime rings, Int. J. Math. Math. Sci. 2004 (2004), no. 37–40, 1957–1964.



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