Strong commutativity and Engel condition preserving maps in prime and semiprime rings

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, L a non-central Lie ideal of R, F and G two non-zero generalized derivations of R. If [F(u), u]G(u) = 0 for all u ∈ L, then one of the following holds: (a) there exists λ ∈ C such that F(x) = λx, for all x ∈ R; (b) R ⊆ M 2 (F), the ring of 2 × 2 matrices over a field F , and there exist a ∈ U and λ ∈ C such that F(x) = ax + xa + λx, for all x ∈ R.

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“…Since z C and by Proposition 2.12 we get w + γz ∈ C and c = 0. Thus by (3.12) it follows that U satisfies 14) and in particular U satisfies…”

Section: The Main Theoremmentioning

confidence: 92%

“…We premit the following result (for the proof see Proposition 1 in [14]): Lemma 2.1. Let F be a field of char(F ) 2, R = M t (F ) the matrix ring over F and t ≥ 3.…”

Section: The Case Of Inner Generalized Derivationsmentioning

confidence: 99%

Posner’s second theorem and some related annihilating conditions on lie ideals

Albaş

Argaç

Vincenzo

2018

Filomat

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“…[11],Proposition 1). Let H be a field of characteristic different from2, R = M t (H) the matrix ring over H and t Let a, b be elements of R, with a = t r,s=1 a rs e rs and b = t r,s=1 b rs e rs , with a rs , b rs ∈ H. For any automorphism ϕ of R, we denote ϕ(a) = t r,s=1 ϕ(a) rs e rs , ϕ(b) = t r,s=1 ϕ(b) rs e rs , with ϕ(a) rs , ϕ(b) rs ∈ H. If a ij b ij = 0 for any i = j and ϕ(a) ij ϕ(b) ij for any i = j and for any ϕ ∈ Aut(R), then a ∈ Z(R) or b ∈ Z(R).…”

mentioning

confidence: 96%

Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings

Filippis

2016

Czech Math J

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“…According to the paper [6] a pair of maps F and G from a ring R into itself is called mutually strong Engel condition preserving (sep) on…”

mentioning

confidence: 99%

“…In [6] both the pair of generalized derivations of a prime ring R being sep on some subset of R and the pair of derivations of a semiprime ring R being sep on a Lie ideal of R were explored.…”

mentioning

confidence: 99%

Nonadditive Strong Commutativity Preserving Derivations and Endomorphisms

Zhang¹,

Xu²

2014

Bulletin of the Korean Mathematical Society

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Abstract. Let S be a nonempty subset of a ring R. A map f : R → R is called strong commutativity preserving on S if [f (x), f (y)] = [x, y] for all x, y ∈ S, where the symbol [x, y] denotes xy − yx. Bell and Daif proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal ρ of R, then ρ ⊆ Z, the center of R. Also they proved that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal I ∪ T −1 (I), then R contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation D and the endomorphism T .In this paper let R always denote an associated ring with the center Z. For a semiprime ring R denote by C and U its extended centroid and maximal right ring of quotients respectively (see [2] for details).Recall that for a nonempty subset S of a ring R, a map f : R → R is called strong commutativity preserving (scp) on S if [f (x), f (y)] = [x, y] for all x, y ∈ S, where the symbol [x, y] denotes xy − yx. The concept of strong commutativity preserving maps was first introduced by Bell and Mason [4]. Bell and Daif [3] proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal ρ of R, then ρ ⊆ Z, the center of R (see also [11,12] due to Liu et al. and [13] by Ma et al. for the results on generalized derivations). In [3] Bell and Daif also obtained that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal I ∪ T −1 (I), then R contains a nonzero central ideal.Brešar and Miers [5] proved that for a semiprime ring R an additive map f : R → R being strong commutativity preserving on R must have the form f (x) = λx + ξ(x), where λ ∈ C, λ 2 = 1, and ξ is an additive map of R into C.

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Strong commutativity and Engel condition preserving maps in prime and semiprime rings

Cited by 51 publications

References 28 publications

Posner’s second theorem and some related annihilating conditions on lie ideals

Posner’s second theorem and some related annihilating conditions on lie ideals

Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings

Nonadditive Strong Commutativity Preserving Derivations and Endomorphisms

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