Narrow Results

Let $R$ be a prime ring of characteristic different from $2$, $Q_r$ be its right Martindale quotient ring and $C$ be its extended centroid, $\alpha$ be an automorphism of $R$, $d$ be a skew derivation of $R$ with associated automorphism $\alpha$, $F$ and $G$ be two nonzero $X$-generalized skew derivation of $R$ with associated term $(b,\alpha ,d)$ and $(b^{\prime },\alpha ,d)$, respectively, $S$ be the set of the evaluations of $f(x_1,\dots ,x_n)$ on $R$, where $f(x_1,\dots ,x_n)$ is a non-central multilinear polynomial over $C$ in $n$ non-commuting variables. Let $0\ne v\in R$ be such that $F(x)x+G(x)xv=0$ for all $x\in S$. Then one of the following statements holds :

• (a)$f{(x_1,\dots ,x_n)}^2$ is central valued on $R$ and there exist $a,a^{\prime }\in Q_r$ such that $F(x)=ax$, $G(x)=a^{\prime }x$ for any $x\in R$ with $a+a^{\prime }v=0$;
• (b)$v\in C$ and $F=-vG$.

In the last part of the paper, we present some applications on the basis of the foregoing proposed result.

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X₁, …,X_t) a multilinear polynomial over K. If g(f (X₁, …,X_t))ⁿ ∈ Z for all x₁, …, x_t ∈ R, then either there exists an element λ ∈ C such that g(x)= λx for all x ∈ R or f(x₁, …,x_t) is central-valued on R except when R satisfies s₄, the standard identity in four variables.

Let R be a prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C. Let g be a generalized derivation of R, f(x₁,…,x_n) a multilinear polynomial over C, a ∈ R, and I a nonzero right ideal of R. Suppose that a[g(f(r₁,…,r_n)), f(r₁,…,r_n)]=0 for all r_i∈ I and aI ≠ 0. Then either g(x)=a₁x with (a₁-γ)I=0 for some a₁∈ U and γ ∈ C, or there exists an idempotent element e ∈soc(RC) such that IC=eRC and one of the following holds: (i) f(x₁,…,x_n) is central-valued in eRe; (ii) g(x)=bx+xc, where b, c ∈ U with (c-b-α)e=0 for some α ∈ C and f(x₁,…,x_n) is central-valued in eRe.