Narrow Results

Let $R$ be a prime ring and $L$ a non-central Lie ideal of $R.$ We aim in this paper to classify generalized derivations $F$ of $R$ satisfying the following algebraic identity:

Given two unital associative rings R ⊆ S, the ring S is said to be an ideal (or Dorroh) extension of R if S = R ⊕ I, for some ideal I ⊆ S. In this note, we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These are generalizations of earlier results in the literature.

We study an associative algebra A over an arbitrary field that is a sum of two subalgebras A₁ and A₂ (i.e. A = A₁ + A₂). Additionally we assume that A_i has an ideal of finite codimension in A_i which satisfies a polynomial identity f_i = 0 for i = 1, 2. Suppose that all rings R = R₁ + R₂, which are sums of subrings R₁ and R₂, are PI rings when R_i satisfies the polynomial identity f_i = 0 for i = 1, 2. We prove that A is a PI algebra.

Let R be a prime ring of characteristic different from 2 with extended centroid C. Let F be a generalized derivation of R, L a non-central Lie ideal of R, f(x₁, …, x_n) a polynomial over C and f(R)={f(r₁, …, r_n): r_i ∈ R}. We study the following cases: (1) [F(u), F(v)]_k=0 for all u, v ∈ L, where k ≥ 1 is a fixed integer; (2) [F(u), F(v)] = 0 for all u, v ∈ f(R); (3) F(u) ◦ F(v)=0 for all u, v ∈ f(R); (4) F(u) ◦ F(v)=u ◦ v for all u, v ∈ f(R). We obtain a description of the structure of R and information on the form of F.

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 K be a commutative ring with unity, R a non-commutative prime K-algebra with center Z(R), U the Utumi quotient ring of R, C=Z(U) the extended centroid of R, I a non-zero two-sided ideal of R, H and G non-zero generalized derivations of R. Suppose that f(x₁,…,x_n) is a non-central multilinear polynomial over K such that H(f(X))f(X)-f(X)G(f(X))=0 for all X=(x₁,…,x_n)∈ Iⁿ. Then one of the following holds: (1) There exists a ∈ U such that H(x)=xa and G(x)=ax for all x ∈ R. (2) f(x₁,…,x_n)² is central valued on R and there exist a, b ∈ U such that H(x)=ax+xb and G(x)=bx+xa for all x ∈ R. (3) char(R)=2 and R satisfies s₄, the standard identity of degree 4.

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, and H, G nonzero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that H(uⁿ)uⁿ + uⁿG(uⁿ) ∈ C for all u ∈ L, then either there exists a ∈ U such that H(x) = xa and G(x) = -ax, or R satisfies the standard identity s₄ and one of the following holds: (i) char(R) = 2; (ii) n is even and there exist a′ ∈ U, α ∈ C and derivations d, δ of R such that H(x) = a′ x + d(x) and G(x) = (α-a′)x + δ(x); (iii) n is even and there exist a′ ∈ U and a derivation δ of R such that H(x)=xa′ and G(x) = -a′ x + δ(x); (iv) n is odd and there exist a′, b′ ∈ U and α, β ∈ C such that H(x) = a′ x + x(β-b′) and G(x) = b′ x+x(α-a′); (v) n is odd and there exist α, β ∈ C and a derivation d of R such that H(x) = α x+d(x) and G(x) = β x + d(x); (vi) n is odd and there exist a′ ∈ U and α ∈ C such that H(x) = xa′ and G(x) = (α - a′)x. As an application of this purely algebraic result, we obtain some range inclusion results of continuous or spectrally bounded generalized derivations H and G on Banach algebras R satisfying the condition H(xⁿ)xⁿ + xⁿG(xⁿ) ∈ rad(R) for all x ∈ R, where rad(R) is the Jacobson radical of R.

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, f(x₁,…,x_n) a multilinear polynomial over C which is not an identity for R, F and G two non-zero generalized derivations of R. If F(u)G(u)=0 for all u ∈ f(R)= {f(r₁,…,r_n): r_i∈ R}, then one of the following holds: (i) There exist a, c ∈ U such that ac=0 and F(x)=xa, G(x)=cx for all x ∈ R; (ii) f(x₁,…,x_n)² is central valued on R and there exist a, c ∈ U such that ac=0 and F(x)=ax, G(x)=xc for all x ∈ R; (iii) f(x₁,…,x_n) is central valued on R and there exist a,b,c,q ∈ U such that (a+b)(c+q)=0 and F(x)=ax+xb, G(x)=cx+xq for all x ∈ R.

Let σ, τ be automorphisms of a ring R. In the present paper many concepts related to biadditive mappings of rings, viz. σ-left centralizer traces, symmetric generalized (σ, τ)-biderivations, σ-left bimultipliers and symmetric generalized Jordan (σ, τ)-biderivations are studied. Many results related to these concepts are given. It is established that every symmetric generalized (σ, τ)-biderivation of a prime ring of characteristic different from 2, can be reduced to a σ-left bimultiplier under certain algebraic conditions. Further, it is shown that every symmetric generalized Jordan (σ, τ)-biderivation of a prime ring of characteristic different from 2 is a symmetric generalized (σ, τ)-biderivation.

Let $R$ be a prime ring of characteristic different from $2$, $Q_r$ be the right Martindale quotient ring of $R$, $C=Z(Q_r)$ the extended centroid of $R$, $L$ be a noncentral Lie ideal of $R$, $H$ and $G$ be two nonzero $b$-generalized derivations of $R$. Suppose there exist fixed integers $m,n\ge 1$ such that $H(u^m)u^n-u^{n}G(u^m)=0$, for all $u\in L$, then either $R$ satisfies the standard identity $s_4(x_1,\dots ,x_4)$ or there is $a^{\prime }\in Q_r$ such that $H(x)=x{a}^{\prime }$, $G(x)=a^{\prime }x$, for any $x\in R$, and one of the following holds:

• ^(a)there exists $a^{\prime }\in C$,
• ^(b)${[x_1,x_2]}^m$ and ${[x_1,x_2]}^n$ are $C$-linearly dependent.

Then, in the second part of the paper we prove a similar result in the case $H$ and $G$ are generalized skew derivations of $R$ such that $H(x^m)x^n-x^{n}G(x^m)=0$, for all $x\in R$.

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).

No abstract received.

Let R be a prime rings and I ≠ 0 an ideal of R. An additive mapping D : R → R is called a left derivation on R if D(xy) = D(x)y + xD(y), holds for all x, y ∈ R. In the present paper, we have discussed the commutativity of a prime rings admitting a left derivation D satisfying several conditions.