Commutativity of near-rings with derivations by using algebraic substructures

Kamal, Ahmed A. M.; Al-Shaalan, Khalid H.

doi:10.1007/s13226-012-0017-0

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…where U is a subsemigroup of (R, •) which provides an interesting extension of many known results (see [5,7,9,10]), where we show that U is commutative, V = {u 2 |u ∈ U } is a subsemigroup of R contained in Z (R) and either 2R = {0} or U ⊆ Z (R). Moreover, we study three cases: nU = {0}, nU = {0}, and R is of characteristic zero.…”

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confidence: 65%

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On some properties of near-rings

Al-Shaalan

2020

Arab. J. Math.

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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 $$d([U,V])\subseteq Z(R)$$ d ( [ U , V ] ) ⊆ Z ( R ) , $$d(U)\subseteq Z(R)$$ d ( U ) ⊆ Z ( R ) , $$x_{o}d(R)\subseteq Z(R)$$ x o d ( R ) ⊆ Z ( R ) , and $$Ux_{o}\subseteq Z(R)$$ U x o ⊆ 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 $$x\in Z(R)$$ x ∈ Z ( R ) , then $$d(x)\in Z(R)$$ d ( x ) ∈ Z ( R ) . In addition, we provide useful examples illustrating our results.

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confidence: 65%

“…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.…”

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confidence: 99%

On some properties of near-rings

Al-Shaalan

2020

Arab. J. Math.

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“…Then we have x k = 0 for some k ∈ N. As N is left Boolean nearring, we have Proof. The proof follows from Corollary 2.2 of Kamal and Al-Shaalan [10] and (1) of Corollary 2.5.…”

Section: Derivationsmentioning

confidence: 81%

“…D(xn) = 0) for all x ∈ R then R is a commutative ring.Proof. The proof follows from Corollary 3.4 (ii) of Kamal and Al-Shaalan[10] and Theorem 2.4. Let R be a 3-prime left(resp.…”

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confidence: 84%

Extensions of Boolean Rings and Nearrings

Nayak¹,

Kuncham

Kedukodi

2019

J. Sib. Fed. Univ., Math. phys.

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In this paper, we introduce the notions of left (right) Boolean rings and nearrings. We give examples to show that left (right) Boolean rings are not commutative in general. We obtain interrelations among these algebraic structures and get conditions under which the structures are commutative. Finally, we study the concept of derivations on left (right) Boolean rings and nearrings and obtain commutativity results.

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Commutativity of rings and near-rings with generalized derivations

Kamal

Al-Shaalan

2013

Indian J Pure Appl Math

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Commutativity of near-rings with derivations by using algebraic substructures

Cited by 4 publications

References 6 publications

On some properties of near-rings

On some properties of near-rings

Extensions of Boolean Rings and Nearrings

Commutativity of rings and near-rings with generalized derivations

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