Semiprime ring
Multiplicative (Generalized) Reverse Derivations on Semiprime Ring
... manner the conclusion can be obtained for the case [F(x), y] − yx ∈ Z(R) for all x, y ∈ I. Using similar technique with some necessary variations, we can prove the following: Theorem 3.9. Let R be a semiprime ...
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On the Lie ring of derivations of a semiprime ring
... Throughout the text R stands for an associative ring (possibly without identity) and n for a positive integer. By Z(R) we denote the center of R. The ring R is called semiprime, if it has no nonzero ...
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A note on a pair of derivations of semiprime rings
... a semiprime ring and prove the ...a semiprime ring R satisfying f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are central (Theorem ...a semiprime ring R is skew-commuting ...
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Lie Ideals and Generalized Derivations in Semiprime Rings
... δ(x)F (x) = 0) for all x ∈ [R, R]. In light of [6], since δ 6= 0 and [R, R] is not central in R, one has the contradiction that F = 0. Lemma 1.3. Let R be a 2-torsion free semiprime ring and L be a ...
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Identities with derivations and automorphisms on semiprime rings
... prime ring, then the ring is commutative (Mayne’s ...prime ring R is of the form x → λx + ζ(x) where λ is an element of C and ζ : R → C is an additive mapping, should be ...2-torsion-free ...
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Dependent Elements of Derivations on Semiprime Rings
... of semiprime rings and on the other hand by the work done by various researchers on commuting derivations on prime and semiprime rings, we investigate some properties, not already investigated, of dependent ...
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An Equation Related to Centralizers in Semiprime Gamma Rings
... of semiprime rings and proved that Jordan centralizers and centralizers of this rings ...free semiprime ring and T : R → R is an additive mapping such that 2 T ( xyx ) = T ( x ) yx + xyx holds for ...
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Some Relations Related to Centralizers on Semiprime Semiring
... on semiprime rings and proved that Jordan centralizer and centralizers of this rings ...free semiprime ring and ܶ: ܴ → ܴ is an additive mapping such that 2ܶ(ݔݕݔ) = ܶ(ݔ)ݕݔ + ݔݕܶ(ݔ) holds for all ݔ, ݕ ...
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On Symmetric Reverse Ɵ*-Bicentralizer in Semiprime *- Rings
... free semiprime *- ring , then every symmetric right Jordan * - Bicentralizer of R is a symmetric right reverse * - Bicentralizer ...*- ring with identity ...
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On Prime and Semiprime Rings with Additive Mappings and Derivations
... We begin our discussion with the following theorem which extends some known results obtained in [8]. Theorem 3.1. Let R be a semiprime ring, I a nonzero ideal of R, and let F, d : R → R be two additive ...
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Remarks on Generalized Derivations in Prime and Semiprime Rings
... a ring with center Z and I a nonzero ideal of ...the semiprime ring R must contains a nonzero central ideal, provided d I / ...prime ring, R must be commutative, provided d / ...
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On commutativity of prime and semiprime rings with generalized derivations
... Fact 2.2. [Lee, 1999, Theorem 4] Let R be a semiprime ring. Then every gen- eralized derivation F on a dense right ideal of R is uniquely extended to U and assumes the form F (x) = ax + d(x) for some a ∈ U ...
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Symmetric Skew Reverse n-Derivations on Prime Rings and Semiprime rings
... Abstract: Let n ≥ 2 be any fixed positive integer and ä denote the trace of symmetric skew reverse n-derivation Ä :R n → R, associated with an antiautomorphism á * .Let I be any Ideal of R.(1)If R is non commutative ...
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On Finite Rank Operators on Centrally Closed Semiprime Rings
... It is also well know that the extended centroid of a prime ring is a field, however, for a semiprime ring, we can only assert that said extended centroid is a von Neumann regular ring. This is ...
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Annihilator condition of a pair of derivations in prime and semiprime rings
... P ROOF : Since any derivation d can be uniquely extended to a derivation in U , and U and R satisfy the same differential identities ( see [20]), then ad(x) ∈ Z(U ), for all x ∈ U . Let P ∈ M(C) such that U/P U is ...
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Transparency of Polynomial Ring Over a Commutative Noetherian Ring
... Proof. R[x ; σ, δ] is Noetherian ring by Hilbert Basis Theorem, namely Theorem (1.12) of Goodearl and Warfield [12]. Now R is a commutative Noetherian Q-algebra, therefore, the ideal { 0 } has a reduced primary ...
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Orthogonal Symmetric Higher bi-Derivations on Semiprime Г-Rings
... free semiprime Г- ring such that 𝑥𝛼𝑦𝛽𝑧 = 𝑥𝛽𝑦𝛼𝑧 for all α, βϵГ , x, y, zϵM and two symmetric higher bi-derivations 𝑑 𝑛 and 𝑔 𝑛 for all n𝜖N are orthogonal if and only if 𝑑 𝑛 (𝑥, 𝑦)𝛼𝑔 𝑛 (𝑦, 𝑧) + 𝑔 𝑛 (𝑥, 𝑦)𝛼𝑑 𝑛 ...
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Effective ring
... b / ∈ J (aR) then by simple observation element r is not a unit in a ring R. Then equality rR + bR = R implies that ru + bv = 1 for some elements u, v ∈ R. Then rsu + bsv = s. Since a = rs then rs = 0 and we have ...
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Phyllostictine A ring assembly via ring closing metathesis
... Finally, ring clo- sure with KOt-Bu at − 15 °C yielded α-methylene-β-lactam 66, in an acceptable 23% yield over the 5 steps (Scheme ...β-lactam ring, the α-methylene functionality also has distinctive ...
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The Largest Left Quotient Ring of a Ring
... Properties of the maximal left quotient rings of a ring. The next theorem describes various properties of the maximal left quotient rings of a ring. In particular, their groups of units and their largest ...
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