arXiv subjects

Jheng-Huei Lin

Publications and source records attributed to Jheng-Huei Lin.

4 recordsLinked to original sources

arXiv2025

Commutators and products of Lie ideals of prime rings

Motivated by some recent results on Lie ideals, it is proved that if $L$ is a Lie ideal of a simple ring $R$ with center $Z(R)$, then $L\subseteq Z(R)$, $L=Z(R)a+Z(R)$ for some noncentral $a\in L$, or $[R, R]\subseteq L$, which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let $R$ be a prime ring with extended centroid $C$. We completely characterize Lie ideals $L$ and elements $a$ of $R$ such that $L+aL$ contains a nonzero ideal of $R$. Given noncentral Lie ideals $K, L$ 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, we characterize noncentral Lie ideals $K_1,\ldots,K_m$ with $m\geq 2$ such that $K_1K_2\cdots K_m$ contains a nonzero ideal of $R$. Finally, we 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.

math.RA↗

arXiv2024

Prime rings having nontrivial centralizers of (skew) traces of Lie ideals

Let $R$ be a prime ring with center $Z(R)$ and with involution $*$. Given an additive subgroup $A$ of $R$, let $T(A):=\{x+x^*\mid x\in A\}$ and $K_0(A):=\{x-x^*\mid x\in A\}$. Let $L$ be a non-abelian Lie ideal of $R$. It is proved that if $d$ is a nonzero derivation of $R$ satisfying $d(T(L))=0$ (resp. $d(K_0(L))=0$), then $T(R)^2\subseteq Z(R)$ (resp. $K_0(R)^2\subseteq Z(R)$). These results are applied to the study of $d(T(M))=0$ and $d(K_0(M))=0$ for noncentral $*$-subrings $M$ of a division ring $R$ such that $M$ is invariant under all inner automorphisms of $R$, and for noncentral additive subgroups $M$ of a prime ring $R$ containing a nontrivial idempotent such that $M$ is invariant under all special inner automorphisms of $R$. The obtained theorems also generalize some recent results on simple artinian rings with involution due to M. Chacron.

math.RA↗

arXiv2024

Certain functional identities on division rings of characteristic two

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 $\text{char}\, D=2$.

math.RA↗

arXiv2024

Certain functional identities on division rings

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 Merchán's result in 2023 to get a complete solution.

math.RA↗