Narrow Results

Let $R$ be a prime ring, let $L$ be a noncentral Lie ideal of $R$ and let $g,h$ be two generalized derivations of $R$. In this paper, we characterize the structure of $R$ and all possible forms of $g$ and $h$ such that ${[g(x^m)x^n-x^{s}h(x^t),x^r]}_k=0$ for all $x\in L$, where $m,n,s,t,r,k$ are fixed positive integers. With this, several known results can be either deduced or generalized. In particular, we give a Lie ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317] and describe a more complete version of the theorem recently obtained by Dhara and De Filippis in [Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48 (2020) 154–167].

Let R be a ring and α,β be endomorphisms of R. An additive mapping F: R → R is called a generalized (α,β)-derivation on R if there exists an (α,β)-derivation d: R → R such that F(xy)=F(x) α(y) + β(x)d(y) holds for all x, y ∈ R. In the present paper, we discuss the commutativity of a prime ring R admitting a generalized (α,β)-derivation F satisfying any one of the properties: (i) [F(x),x]_α,β=0, (ii) F([x,y])=0, (iii) F(x ◦ y)=0, (iv) F([x,y])=[x,y]_α,β, (v) F(x ◦ y)=(x ◦ y)_α,β, (vi) F(xy)- α(xy) ∈ Z(R), (vii) F(x)F(y)- α(xy) ∈ Z(R) for all x, y in an appropriate subset of R.

Let R be a semiprime nonassociative ring satisfying (x, y, z)–(z, y, x) ∈ N_r then N_l = N_r where N_l and N_r are Lie ideals of R, the set {x ∈ N_r : (R, R, R)x = 0} = {x ∈ N_l : x(R, R, R) = 0} is an ideal of R, and it is contained in the nucleus. Further if [R, R]N_r ⊂ N_r and R is a prime ring with N_r ≠ 0 then R is either associative or commutative.

Let $R$ be a prime ring of characteristic not equal to $2$, $U$ be the Utumi quotient ring of $R$ and $C$ be the extended centroid of $R$. Let $G$ and $F$ be two generalized derivations on $R$ and $L$ be a non-central Lie ideal of $R$. If $F(G(u))u=uG(u)$, for all $u\in L$, then one of the following holds :

• ⁽¹⁾$G=0$.
• ⁽²⁾there exists $\lambda \in C$ such that $F(x)=x$, and $G(x)=\lambda x$ for all $x\in R$.
• ⁽³⁾$R$ satisfies the standard identity $s_4$.

Let $R$ be a semiprime ring, $U$ a square-closed Lie ideal of $R$ and $D:R\times R\to R$ a symmetric bi-derivation and $d$ be the trace of $D.$ In this paper, we shall prove that $R$ contains a nonzero central ideal if any one of the following holds: (i) $d([x,y])\pm [x,y]\in Z,$ (ii) $[d(x),d(y)]\pm [x,y]\in Z,$ (iii) $d(x\circ y)\pm x\circ y\in Z,$ (iv) $d(x)\circ d(y)\pm x\circ y\in Z,$ (v) $d(x\circ y)\pm [x,y]\in Z,$ (vi) $d(x)\circ d(y)\pm [x,y]\in Z,$ (vii) $d([x,y])\pm x\circ y\in Z,$ (viii) $[d(x),d(y)]\pm x\circ y\in Z,$ (ix) $d(x)d(y)\pm xy\in Z,$ (x) $d(x)d(y)\pm yx\in Z,$ (xi) $d(x)d(y)\pm [x,y]\in Z$ and (xii) $d(x)d(y)\pm x\circ y\in Z$, for all $x,y\in U.$

Let $R$ be an $n!$-torsion free semiprime ring with center $Z(R)$ and $D,G:R^n\to R$ be two $n$-additive mappings with traces $d,g:R\to R$, respectively. Sǒgǔtchǔ and Gǒlbasi [E. K. Sǒgǔtchǔ and Ǒ. Gǒlbasi, Commutativity theorems on Lie ideals with symmetric bi-derivations in semiprime rings, Asian Eur. J. Math. 16(7) (2023) 2350129] studied the following identities for symmetric bi-derivations:

• ⁽ⁱ⁾$d([x,y])\pm [x,y]\in Z(R)$,
• ⁽ⁱⁱ⁾$[d(x),g(y)]\pm [x,y]\in Z(R)$,
• ⁽ⁱⁱⁱ⁾$d(x\circ y)\pm (x\circ y)\in Z(R)$,
• ^(iv)$d(x)\circ g(y)\pm (x\circ y)\in Z(R)$,
• ^(v)$d(x\circ y)\pm [x,y]\in Z(R)$,
• ^(vi)$d(x)\circ g(y)\pm [x,y]\in Z(R)$,
• ^(vii)$d([x,y])\pm (x\circ y)\in Z(R)$,
• ^(viii)$[d(x),g(y)]\pm (x\circ y)\in Z(R)$,
• ^(ix)$d(x)g(y)\pm xy\in Z(R)$,
• ^(x)$d(x)g(y)\pm yx\in Z(R)$,
• ^(xi)$d(x)g(y)\pm [x,y]\in Z(R)$,
• ^(xii)$d(x)g(y)\pm (x\circ y)\in Z(R)$,

for all $x,y\in U$, where $U$ is a square closed Lie ideal of $R$ and then obtained that $R$ contains a nonzero central ideal. In this paper, we prove that the conclusion of above results holds for trace of any $n$-additive mapping (not necessarily to be symmetric bi-derivation).

Let R be a ring with centre Z(R). A biadditive symmetric mapping D(., .) : R × R → R is called symmetric biderivation if for any fixed y ∈ R, the mapping x ↦ D(x, y) is a derivation. A mapping f : R → R defined by f(x) = D(x, x) is called the trace of D. In this paper we prove that a nonzero Lie ideal L of a semiprime ring R of characteristic different from two is central if it satisfies any one of the following properties: (i) f(xy) ∓ [x, y] ∈ Z(R), (ii) f(xy) ∓ [y, x] ∈ Z(R), (iii) f(xy) ∓ xy ∈ Z(R), (iv) f(xy) ∓ yx ∈ Z(R), (v) f([x, y]) ∓ [x, y] ∈ Z(R), (vi) f([x, y]) ∓ [y, x] ∈ Z(R), (vii) f([x, y]) ∓ xy ∈ Z(R), (viii) f([x, y]) ∓ yx ∈ Z(R), (ix) f(xy) ∓ f(x) ∓ [x, y] ∈ Z(R), (x) f(xy) ∓ f(y) ∓ [x, y] ∈ Z(R), (xi) f([x, y]) ∓ f(x) ∓ [y, x] ∈ Z(R), (xii) f([x, y]) ∓ f(x) ∓ [y, x] ∈ Z(R), (xiii) f([x, y]) ∓ f(y) ∓ [x, y] ∈ Z(R), (xiv) f([x, y]) ∓ f(y) ∓ [y, x] ∈ Z(R), (xv) f([x, y]) ∓ f(xy) ∓ [x, y] ∈ Z(R), (xvi) f([x, y]) ∓ f(xy) ∓ [y, x] ∈ Z(R), (xvii) f(x)f(y) ∓ [x, y] ∈ Z(R), (xviii) f(x)f(y) ∓ [y, x] ∈ Z(R), (xix) f(x)f(y)∓xy ∈ Z(R), (xx) f(x)f(y)∓yx ∈ Z(R), where f stands for the trace of a biadditive symmetric mapping D(., .) : R×R → R. Moreover, motivated by a well known theorem of Posner [11, Theorem 2] and a result of Deng and Bell [6, Theorem 2], we prove that if R admits a symmetric biderivation D such that the trace f of D is n-centralizing on L, then f is n-commuting on L.