LIE IDEALS, MORITA CONTEXT AND GENERALIZED (α, β)-DERIVATIONS

A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson＇s famous result, several tech- niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized （α β）-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.

An Identity with Skew Derivations on Lie Ideals

Let R be a 2-torsion free prime ring and L a noncommutative Lie ideal of R. Suppose that （d,σ） is a skew derivation of R such that xsd（x）xt = 0 for all x ∈ L, where s, t are fixed non-negative integers. Then d = 0.

Left Annihilator of Power Values with Derivations on Lie Ideals in Prime Rings

Let R be a prime ring with characteristic di erent from two,d a derivation of R,L a noncentral Lie ideal of R,and a2R.In the present paper it is showed that if a(d(u^m)±u^m)^n for all u∈L,where m;n are xed positive integers,then a=0 unless R satis es s4,the standard polynomial identity in four variables.

Centralizing Automorphisms of Lie Ideals in Prime Rings of Characteristic 2

Let R be a prime ring with an automorphism σ≠1, an identity map. Let L be a noncentral Lie ideal of R such that \xσ, x] ∈Z for all x ∈ L, where Z is the center of R. Then L is contained in the center of R, unless char(R) = 2 and dimcRC = 4.

Generalized Fuzzy Lie Ideals of Lie Algebras

In this paper we introduce new generalized fuzzy Lie ideals of Lie algebras and study some of their important properties.We characterize these generalized Lie ideals of Lie algebras by their level subsets.Some characterization of the generalized fuzzy Lie ideals of Lie algebras are also established.

Lie Ideals in AF C~*-Algebras

We study Lie ideals in unital AF C^＊-algebras. It is shown that if a linear manifold L in an AF C^＊-algebra A is a closed Lie ideal in A, then there exists a closed associative ideal I and a closed subalgebra EI of the canonical masa D of A such that [A,I]^- belong to L belong to I ＋ EI, and that every closed subspace in this form is a closed Lie ideal in A.

A Note on Commutators with Power Central Values on Lie Ideals

This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identity in 4 variables. The case where R is a semiprime ring is also examined by the authors. The results of the note improve Carini and Filippis＇s results.

Derivations as Homomorphisms or Anti-homomorphisms on Lie Ideals

Let R be a 2-torsion free prime ring, Z the center of R, and U a nonzero Lie ideal of R. If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U, then either d = 0 or U lohtein in Z. This result improves a theorem of Asma, Rehman, and Shakir.

Certain Additive Maps on （m ＋ n ＋ 1）-Power Closed Lie Ideals

Let R be a prime ring and m, n be fixed non-negative integers such that m＋n ≠ 0. Suppose L is an （m＋m＋1）-power closed Lie ideal, and this means ure＋n＋1 ∈ L for all u ∈ L. If charR = 0 or a prime p 〉 2（m ＋ n）, we characterize the additive maps d： L → R satisfying d（um＋n＋1） = （m -＋n ＋ 1）umd（u）un （resp., d（um＋n＋l） = umd（u）un） for all u ∈ L.

Two Results on Square Closed Lie Ideals of Prime Rings

Let R be a 2-torsion free prime ring, d1 a nonzero derivation, -γ a generalized derivation associated with a nonzero derivation d2, U a square closed Lie ideal of R. In the present paper,we prove that if [di^2（u）, u] ∈ Z（R） or γ acts as a homomorphism （or an antihomomorphism） on U, then U Z（R）.

Some Conditions of Central Lie Ideal

Let di（1≤ i≤n）, 51,52, 53 be nonzero derivations of a prime ring R with char R ≠ 2. Suppose that U is a Lie ideal such that u2 ∈ U for all u ∈ U. In this paper, we prove that U [U→] Z（R） when one of the following holds： （1） d1（x1）d2（x2）,… ,dn（xn）∈Z（R） （2） δ3（y）δ1（x） = δ2（x）δ3（y）. Further, if g is a Lie ideal and a subring then （3） δ1（x）δ2（y） ＋δ2（x）δ1（y） ∈ Z（R） for all xi,x,y ∈ U.

Notes on Automorphisms of Prime Rings

Let R be a prime ring, L a noncentral Lie ideal and a nontrivialautomorphism of R such that us（u）ut = 0 for all u 2 L, where s; t are fixednon-negative integers. If either charR 〉 s ＋ t or charR = 0, then R satisfies s4, thestandard identity in four variables. We also examine the identity （σ（[x; y]）-[x; y]）n =0 for all x; y ∈ I, where I is a nonzero ideal of R and n is a fixed positive integer. Ifeither charR 〉 n or charR = 0, then R is commutative.

Prime Rings with Generalized Derivations

The concept of derivations and generalized inner derivations has been generalized as an additive function δ： R → R satisfying δ （xy） = δ（x）y＋xd（y） for all x, y ∈ R, where d is a derivation on R. Such a function δ is called a generalized derivation. Suppose that U is a Lie ideal of R such that u^2 ∈ U for all u ∈ U. In this paper, we prove that U lahtain in Z（R） when one of the following holds： （1） δ（[u, v]） = u o v =（2） δ（[u,v]）=[u o v] = 0 （3） δ（u o v） = [u, v] （4） δ（u o v）＋δ[u, v] = 0 for all u, v ∈ U.