On some properties of near-rings
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Arabian Journal of Mathematics
R E S E A R C H A RT I C L E
Khalid H. Al-Shaalan
On some properties of near-rings
Received: 11 January 2020 / Accepted: 3 November 2020 © The Author(s) 2020
Abstract We establish sufficient conditions for 3-prime near-rings to be commutative rings. In particular, for a 3-prime near-ring R with a derivation d, we investigate conditions such as d([U, V ]) ⊆ Z (R), d(U ) ⊆ Z (R), xo d(R) ⊆ Z (R), and U xo ⊆ Z (R). As a by-product, we generalize and extend known results related to rings and near-rings. Furthermore, we discuss the converse of a well-known result in rings and near-rings, namely: if x ∈ Z (R), then d(x) ∈ Z (R). In addition, we provide useful examples illustrating our results. Mathematics Subject Classification
16N60 · 16W25 · 16Y30
1 Introduction Throughout this paper, R denotes a left near-ring and Z (R) denotes the multiplicative center of R. A near-ring R is called 3-prime if, for all x, y ∈ R, x Ry = {0} implies x = 0 or y = 0. A map d : R → R is a derivation on R if d is an additive mapping and d(x y) = xd(y) + d(x)y for all x, y ∈ R. If the map d : R → R defined by d(x) = xr − r x or by d(x) = r x − xr for all x ∈ R, where r ∈ R, is a derivation (and it will be if R is a ring), then it is called an inner derivation on R induced by r . An element x ∈ R is called a right (left) zero divisor in R if there exists a non-zero element y ∈ R, such that yx = 0 (x y = 0). A zero divisor is a right or a left zero divisor. A near-ring R is called a zero-symmetric near-ring, if 0x = 0 for all x ∈ R. For subsets X, Y ⊆ R, the symbol [X, Y ] denotes the set {x y − yx|x ∈ X, y ∈ Y } . We say that U is a semigroup right (left) ideal of a near-ring R if U is a non-empty subset of R satisfying U R ⊆ U (RU ⊆ U ). We say that U is a semigroup ideal if it is both a semigroup right and left ideal. For further information about near-rings, see [12,13]. Studying when a near-ring is a ring or a commutative ring is one of the most important questions in the theory of near-rings, which has been the subject of many investigations since the 70s of the foregoing century. For example, among many others, Ligh in [11] and Bell in [2–4] have established some conditions that force a near-ring to be a ring. Later on, Bell and Mason in [7] have studied many properties of 3-prime near-rings that make them commutative rings. Beidar, Fong, and Wang [1] have generalized some results about rings to the setting of near-rings, while Wang [15] has established some beautiful properties of near-rings. On the other hand, Bell in [5] has generalized several results of [7] using one (two) sided semigroup ideals of near-rings. Subsequently, many authors, for instance in [8,9] and [10], have investigated various properties of near-rings that make of them commutative rings. In this paper, we establish some new results in this direction. In Sect. 3, we begin with Theorem 3.1 which is a generalization of Theorem 2.3 of [6] about prime rings that satisfy d([x, y]) ∈ Z (R) for all x, y ∈ R. K. H. Al-Shaalan (B) Colle
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