Lie Ideals Research Articles

The orthogonal Lie algebra of operators: Ideals and derivations

We study in this paper the infinite-dimensional orthogonal Lie algebra OC which consists of all bounded linear operators T on a separable, infinite-dimensional, complex Hilbert space H satisfying CTC=−T⁎, where C is a conjugation on H. By employing results from the theory of complex symmetric operators and skew-symmetric operators, we determine the Lie ideals of OC and their dual spaces. We study derivations of OC and determine their spectra. These results complete some results of P. de la Harpe and provide interesting contrasts between OC and the algebra B(H) of all bounded linear operators on H.

Power Values of Generalized Skew Derivations with Annihilator Conditions on Lie Ideals

Let R be a prime ring with maximal right ring of quotients Q, extended centroid C and a noncommutative Lie ideal L. Our aim is to characterize the forms of generalized skew derivations $$\delta $$ and $$\varrho $$ of R in case of $$\begin{aligned} p(\delta (u)^{l_{1}}\varrho (u)^{l_{2}}\delta (u)^{l_{3}}\varrho (u)^{l_{4}}\cdots \varrho (u)^{l_{k}} )^{n}=0, \end{aligned}$$ for all $$u\in L$$ , where $$0 \ne p\in R$$ , $$l_{1},l_{2},\ldots ,l_{k}$$ are fixed nonnegative integers with $$l_{1}\ne 0$$ and n is a positive fixed integer. The proof of this main result is based on the so-called extended Jacobson density theorem, which is different from that of previous results in the literature. Also as a result, we obtain if $$p\delta (u)^{n}=0$$ for all $$u \in L$$ , then there exists $$a\in Q$$ such that $$\delta (x)=ax$$ for all $$x \in R$$ and $$pa=0$$ unless R satisfies $$s_4$$ .

Lie ideals and Jordan triple (α,β)-derivations in rings

In this paper we prove that on a 2-torsion free semiprime ring R every Jordan triple (α,β)-derivation (resp. generalized Jordan triple (α,β)-derivation) on Lie ideal L is an (α,β)-derivation on L (resp. generalized (α,β)-derivation on L)

Nilpotent fuzzy lie ideals

In this paper, we introduce a new definition for nilpotent fuzzy Lie ideal, which is a well-defined extension of nilpotent Lie ideal in Lie algebras, and we name it a good nilpotent fuzzy Lie ideal. Then we prove that a Lie algebra is nilpotent if and only if any fuzzy Lie ideal of it, is a good nilpotent fuzzy Lie ideal. In particular, we construct a nilpotent Lie algebra via a good nilpotent fuzzy Lie ideal. Also, we prove that with some conditions, every good nilpotent fuzzy Lie ideal is finite. Finally, we define an Engel fuzzy Lie ideal, and we show that every Engel fuzzy Lie ideal of a finite Lie algebra is a good nilpotent fuzzy Lie ideal. We think that these notions could be useful to solve some problems of Lie algebras with nilpotent fuzzy Lie ideals.

On Lie Ideals and Automorphisms in Prime Rings

Let R be a prime ring of characteristic different from 2 with center Z and extended centroid C, and let L be a Lie ideal of R. Consider two nontrivial automorphisms α and β of R for which there exist integers m,n ≥ 1 such that α(u)n + β(u)m = 0 for all u ∈ L. It is shown that, under these assumptions, either L is central or R ⊆ M2(C) (where M2(C) is the ring of 2 × 2 matrices over C), L is commutative, and u2 ∈ Z for all u ∈ L. In particular, if L =[R, R], then R is commutative.

On an identity involving generalized derivations and Lie ideals of prime rings

Let $R$ be a prime ring, $U$ the Utumi quotient ring of $R, $ $C$ the extended centroid of $R$ and $L$ a noncentral Lie ideal of $R.$ If $R$admits a generalized derivation $F$ associated with a derivation $\delta$ of $R$ such that for some fixed integers $m, n\geq 1, $ $F([u, v])^{m} = [u, v]_{n}$ for all $u, v\in L, $ then one of the following holds true: (ⅰ) $R$ satisfies $s_{4}, $ the standard identity in four variables. (ⅱ) there exists $\lambda\in C$ such that $F(x) = \lambda x$ for all $x\in R.$ Moreover, if $n = 1, $ then $\lambda^{m} = 1$ and if $n > 1, $ then $F = 0.$

On closed Lie ideals of certain tensor products of C∗‐algebras II

AbstractWe identify all closed Lie ideals of and , where is either the Haagerup tensor product, the Banach space projective tensor product or the operator space projective tensor product, A is any simple ‐algebra, B is any ‐algebra with one of them admitting no tracial states, and H is an infinite dimensional separable Hilbert space. Further, generalizing a result of Marcoux, we also identify all closed Lie ideals of , where A is a simple ‐algebra with at most one tracial state and B is any commutative ‐algebra.

Generalized Skew Derivations and Nilpotent Values on Lie Ideals

Let R be a prime ring of characteristic different from 2 and 3, Qr be its right Martindale quotient ring and C be its extended centroid. Suppose that F and G are generalized skew derivations of R, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Under appropriate conditions we prove that if (F(x)x – xG(x))n = 0 for all x ∈ L, then one of the following holds: (a) there exists c ∈ Qr such that F(x) = xc and G(x) = cx; (b) R satisfies s4 and there exist a, b, c ∈ Qr such that F(x) = ax + xc, G(x) = cx + xb and (a − b)2 = 0.

Generalized skew-derivations acting on Lie ideals with central values in prime rings

Let R be a prime ring with characteristic different from 2 and 3, $$Q_r$$ be its right Martindale quotient ring, C be its extended centroid, F a nonzero generalized skew derivation of R and L a noncentral Lie ideal of R. If $$\begin{aligned}{}[[F(u),u],F(u)]\in Z(R) \end{aligned}$$for all $$u\in L$$, then one of the following holds:

A GENERALIZED STUDY ON CLOSED LIE IDEALS WITH (α,α)-DERIVATIONS

In this paper, we study square closed Lie ideals of semi-prime rings with generalized derivations and investigate commutative properties of square closed Lie ideals under different conditions. Also, we take generalized derivation with determined derivation on prime ring and prove that is commuting on Lie ideal. Finally, we reach the corollaries about commutativity of prime rings by using the theorems we prove.

Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings

Let R be a noncommutative prime ring, I a nonzero left ideal of R, L a non-central Lie ideal of R, U the left Utumi quotient ring of R and the extended centroid of R. Let G be a generalized derivation of R and g be a derivation of R and fixed positive integers. In the present paper, we describe the structure of R and all possible forms of G and g in the following cases: for all and for all

Lie Ideals on Prime Γ-Rings with Jordan Right Derivations

Lie Ideals on Prime Γ-Rings with Jordan Right Derivations

On Skew Left n-Derivations with Lie Ideal Structure

In this paper the centralizing and commuting concerning skew left -derivations and skew left -derivations associated with antiautomorphism on prime and semiprime rings were studied and the commutativity of Lie ideal under certain conditions were proved.

On Skew Left n-Derivations with Lie Ideal Structure

In this paper the centralizing and commuting concerning skew left -derivations and skew left -derivations associated with antiautomorphism on prime and semiprime rings were studied and the commutativity of Lie ideal under certain conditions were proved.

On Certain Identities with Automorphisms on Lie Ideals in Prime and Semiprime Rings

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]t – [x, y]m [α([x, y]),[x, y]]n [x, y]s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s4, the standard identity of degree 4. We also extend the results to semiprime rings.

A note on Lie ideals of prime rings

In this paper, we characterize Lie ideals, which are either finitely generated [Formula: see text]-modules or maximal, in (centrally closed) prime rings. As consequences, we extend the results proved in [1] for finite dimensional central division algebras of characteristic not [Formula: see text] to simple rings of arbitrary characteristic.

Derivations with annihilator conditions on Lie ideals in prime rings

Let [Formula: see text] be a prime ring with characteristic different from two, [Formula: see text] a derivation of [Formula: see text], [Formula: see text] a noncentral Lie ideal of [Formula: see text], and [Formula: see text]. In the present paper, it is shown that if one of the following conditions holds: (i) [Formula: see text], (ii) [Formula: see text], (iii) [Formula: see text] and (iv) [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then [Formula: see text] unless [Formula: see text] satisfies [Formula: see text], the standard polynomial identity in four variables.

Derivations satisfying certain algebraic identities on Lie ideals

Derivations satisfying certain algebraic identities on Lie ideals

On Lie ideals and symmetric generalized (α, β)-biderivation in prime ring

On Lie ideals and symmetric generalized (α, β)-biderivation in prime ring

Strong commutativity preserving derivations on Lie ideals of prime Γ-rings

Strong commutativity preserving derivations on Lie ideals of prime Γ-rings