Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m,n$ fixed positive integers. (i) If ${\left(d[x,y]\right)}^m={[x,y]}_n$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathrm{Char}R\ne 2$ and ${[d\left(x\right),d\left(y\right)]}_m={[x,y]}^n$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.

Let $R$ be a prime ring of char $R\ne 2$ with a nonzero derivation $d$ and let $U$ be its noncentral Lie ideal. If for some fixed integers $n_1\ge 0,n_2\ge 0,n_3\ge 0$, ${\left(u^{n_1}[d\left(u\right),u]u^{n_2}\right)}^{n_3}\in Z\left(R\right)$ for all $u\in U$, then $R$ satisfies $S_4$, the standard identity in four variables.

Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z\left(U\right)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $\left[F\right(u),u]F\left(u\right)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F\left(x\right)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity $s_4$ and there exist $a\in U$ and $\alpha \in C$ such that $F\left(x\right)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion...

Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $\sigma$ be an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$. Then $R$ is said to be an almost $\delta$-divided ring if every minimal prime ideal of $R$ is $\delta$-divided. Let $R$ be a Noetherian ring which is also an algebra over $\mathbb{Q}$ ($\mathbb{Q}$ is the field of rational numbers). Let $\sigma$ be an automorphism of $R$ such that $R$ is a $\sigma (*)$-ring and $\delta$ a $\sigma$-derivation of $R$ such that $\sigma \left(\delta \right(a\left)\right)=\delta \left(\sigma \right(a\left)\right)$ for all $a\in R$. Further,...