Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

On functional identities in left ideals of prime rings

In the present paper we describe surjective Lie and Jordan maps onto left ideals of prime noncommutative rings. Further, we describe bijective linear maps of left ideals of centrally closed prime algebras preserving commutativity.     

Some commutativity criteria for prime rings with differential identities on Jordan ideals

In this article, we present some commutativity theorems for a ring R equipped with a generalized derivation satisfying certain differential identities on Jordan ideals of R. Some related results for prime rings are also discussed. Finally; we provide examples to show that the assumed restrictions are not superfluous.     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

An Engel condition with skew derivations for one-sided ideals

We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and \(\delta \) a nonzero \(\sigma \)-derivation of A, where \(\sigma \) is an epimorphism of A. For \(x,y\in A\), we set \([x,y] = xy - yx\). If \([[\ldots [[\delta (x^{n_0}),x^{n_1}],x^{n_{2}}],\ldots ],x^{n_k}]=0\) for all \(x\in R\), where \(n_{0},n_{1},\ldots ,n_{k}\) are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) \(C\cong GF(2)\), the Galois field of two elements; (3) there exist \(b\in Q\) and \(\lambda \in C\) such that \(\delta (x)=\sigma (x)b-bx\) for all \(x\in A\), \((b-\lambda )R=0\) and \(\sigma (R)=0\). The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).     

Commuting and centralizing generalized derivations on lie ideals in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] _k x ? x[G(x), x] _k = 0 for any x ∈ L, then one of the following holds:

1. either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R
2. or R satisfies the standard identity s ₄(x ₁, …, x ₄) and one of the following conclusions occurs

1. there exist a, b, c, q ∈ U, such that a ?b + c ?q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R
2. there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc?d(x) for all x ∈ R, with a + b ? c ∈ C.

     

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

Let R be a prime ring of characteristic different from 2 and 3, Q_r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y ∈ R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y ∈ R.     

Annihilators of derivations with Engel conditions on lie ideals

LetR a prime ring,L a noncentral Lie ideal ofR anda ∈R. Suppose thatd is a nonzero derivation ofR such thata[d(u),u] _k =0 for allu ∈L, wherek is a fixed positive integer. Thena=0 except when charR=2 and dim _C RC=4.     

High order derivations and primary ideals to regular prime ideals

Ohne Zusammenfassung     

Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals

Let $R$ be a non-commutative prime ring, with center $Z(R)$ , extended centroid $C$ and let $F$ be a non-zero generalized derivation of $R$ . Denote by $L$ a non-central Lie ideal of $R$ . If there exists $0\ne a\in R$ such that $a[F(x),x]_k\in Z(R)$ for all $x\in L$ , where $k$ is a fixed integer, then one of the followings holds: (1) either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$ , (2) or $R$ satisfies $s_4$ , the standard identity in $4$ variables, and $char(R)=2$ ; (3) or $R$ satisfies $s_4$ and there exist $q\in U, \gamma \in C$ such that $F(x)=qx+xq+\gamma x$ .