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In this paper we study commutators and products of noncentral Lie ideals of prime rings. Precisely, let $R$ be a prime ring with extended centroid $C$, and let $K, L$ be noncentral Lie ideals of $R$. It is proved that $[K, L]=0$ if and only if $KC=LC=Ca+C$ for any noncentral element $a\in L$. As a consequence, given noncentral Lie ideals $K_1,\ldots,K_m$ with $m\geq 2$, we prove that $K_1K_2\cdots K_m$ contains a nonzero ideal of $R$ except when, for $1\leq j\leq m$, $K_1C=K_jC=Ca+C$ for any noncentral element $a\in K_1$. Moreover, we also characterize noncentral Lie ideals $K_j$'s and $L_k$'s satisfying $\big[K_1K_2\cdots K_m, L_1L_2\cdots L_n\big]=0$ from the viewpoint of centralizers.

We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral. For a unital algebra over a field of characteristic $\neq 2$ where every additive commutator is a sum of square-zero elements, we show that a fully noncentral subspace is a Lie ideal if and only if it is invariant under all inner automorphisms. This applies in particular to zero-product balanced algebras., Comment: 19 pages

Let $D$ be a noncommutative division ring. In a recent paper, Lee and Lin proved that if $\text{char}\, D\ne 2$, the only solution of additive maps $f, g$ on $D$ satisfying the identity $f(x) = x^n g(x^{-1})$ on $D\setminus \{0\}$ with $n\ne 2$ a positive integer is the trivial case, that is, $f=0$ and $g=0$. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer $n$ if $\text{char}\, D=2$.

For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal $L$ of a ring $R$, we offer a criterion for $R$ to be $L$-semiprime. For a prime ring $R$, we characterizes Lie ideals $L$ of $R$ such that $R$ is $L$-semiprime. Moreover, $X$-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied., Comment: Comments welcome

We study the functional identity $G(x)f(x)=H(x)$ on a division ring $D$, where $f \colon D\to D$ is an additive map and $G(X)\ne 0, H(X)$ are generalized polynomials in the variable $X$ with coefficients in $D$. Precisely, it is proved that either $D$ is finite-dimensional over its center or $f$ is an elementary operator. Applying the result and its consequences, we prove that if $D$ is a noncommutative division ring of characteristic not $2$, then the only solution of additive maps $f, g$ on $D$ satisfying the identity $f(x) = x^n g(x^{-1})$ with $n\ne 2$ a positive integer is the trivial case, that is, $f=0$ and $g=0$. This extends Catalano and Merch\'{a}n's result in 2023 to get a complete solution.

Let R be an associative ring.In the paper we study n-generalized commutators of rings and prove that if R is a noncommutative prime ring and n > 2, then every nonzero n-generalized Lie ideal of R contains a nonzero ideal. Therefore, if R is a noncommutative simple ring, then R = [R, . . . ,R]n. This extends a classical result due to Herstein (Portugal. Math., 1954). Some generalizations and related questions on n-generalized commutators and their relationship with noncommutative polynomials are also discussed., Comment: 35 pages