Narrow Results

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.

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:

Let R be a ring. If we replace the original associative product of R with their canonic Lie product, or [a, b] = ab - ba for every a, b in R, then R would be a Lie ring. With this new product the additive commutator subgroup of R or [R, R] is a Lie subring of R. Herstein has shown that in a simple ring R with characteristic unequal to 2, any Lie ideal of R either is contained in Z(R), the center of R or contains [R, R]. He also showed that in this situation the Lie ring [R, R]/Z[R, R] is simple. We give an alternative matrix proof of these results for the special case of simple artinian rings and show that in this case the characteristic condition can be more restricted.

Cartan–Brauer–Hua Theorem is a well-known theorem which states that if $R$ is a subdivision ring of a division ring $D$ which is invariant under all elements of $D$ or $dR{d}^{-1}\subseteq R$ for all $d\in D\setminus \{0\}$, then either $R=D$ or $R$ is contained in the center of $D$. The invariance idea of this basic theorem is the main notion of this paper. We prove that if $D$ is a division ring with involution ${}^{\ast }$ and $M$ is a subspace of $D$ which is invariant under all symmetric elements of $D$, then either $M$ is contained in the center of $D$ or is a Lie ideal of $D$. Also, we show that if $T$ is a self-invariant subfield of a non-commutative division ring $D$ with a nontrivial automorphism, then $T$ contains at least one non-central proper subfield of $D$.

We show that if $I$ is a non-central Lie ideal of a ring $R$ with Char$(R)\ne 2$, such that all of its nonzero elements are invertible, then $R$ is a division ring. We prove that if $R$ is an $F$-central algebra and $I$ is a Lie ideal without zero divisor such that the set of multiplicative cosets $\{aF|a\in I\}$ is of finite cardinality, then either $R$ is a field or $I$ is central. We show the only non-central Lie ideal without zero divisor of a non-commutative central $F$-algebra $R$ with Char$(R)\ne 2$ and radical over the center is $[R,R]$, the additive commutator subgroup of $R$ and in this case $R$ is a generalized quaternion algebra. Finally we prove that if $I$ is a Lie ideal without zero divisor in a central $F$-algebra with characteristic not 2 and if $(\frac{I+F}{F},+)$ is a finite residual group, then $I$ is central.

A Lie ideal of a division ring $A$ is an additive subgroup $L$ of $A$ such that the Lie product $[l,a]=la-al$ of any two elements $l\in L,a\in A$ is in $L$ or $[l,a]\in L$. The main concern of this paper is to present some properties of Lie ideals of $A$ which may be interpreted as being dual to known properties of normal subgroups of $A^{\ast }$. In particular, we prove that if $A$ is a finite-dimensional division algebra with center $F$ and $\mathrm{\text{char}}F\ne 2$, then any finitely generated $\mathbb{Z}$-module Lie ideal of $A$ is central. We also show that the additive commutator subgroup $[A,A]$ of $A$ is not a finitely generated $\mathbb{Z}$-module. Some other results about maximal additive subgroups of $A$ and $M_n(A)$ are also presented.

Let $D$ be a division algebra over an infinite field $K$ such that every element of $D$ is a sum of finitely many algebraic elements. As a generalization of Asano’s theorem, it is proved that every noncentral subspace of $D$ invariant under all inner automorphisms induced by algebraic elements contains $[D,D]$, the additive subgroup of $D$ generated by all additive commutators of $D$. From the viewpoint we study the existence of normal bases of certain subspaces of division algebras. It is proved among other things that $D$ is generated by multiplicative commutators as a vector space over the center of $D$.

Let $R$ be a prime ring with characteristic different from two, $d$ a derivation of $R$, $L$ a noncentral Lie ideal of $R$, and $a\in R$. In the present paper, it is shown that if one of the following conditions holds: (i) $a{(d{(uv)}^s-{(uv)}^t)}^n=0$, (ii) $a{(d{(uv)}^s+{(uv)}^t)}^n=0$, (iii) $a{(d{(uv)}^s-{(vu)}^t)}^n=0$ and (iv) $a{(d{(uv)}^s+{(vu)}^t)}^n=0$ for all $u,v\in L$, where $n,s,t$ are fixed positive integers, then $a=0$ unless $R$ satisfies $s_4$, the standard polynomial identity in four variables.

For a subset $T$ of a ring $R$ we denote by ${ℐ}(T)$ the ideal of $R$ generated by $T$. Given a higher commutator $L$ of $R$, if $R={ℐ}(L)$ then $R=L+L^2$? The question is motivated by the result that a ring $R$ is equal to its subring generated by $[R,R]$ if $R$ is either a noncommutative simple ring (by Herstein) or a unital ring with $1\in [R,R]$ (by Eroǧlu). In this note, we study the question for the rings $R$ satisfying the property that every proper ideal of $R$ is contained in a maximal ideal (in particular, if $R$ is finitely generated as an ideal).

Let $R$ be a prime ring with the extended centroid $C$ and $L$ a noncentral Lie ideal of $R$. Suppose that $a\in R$, $\gamma \in C$ and $d$ is a derivation of $R$ such that $a\left(\right.d{(u^2)}^s-\gamma u^{2t}{\left)\right.}^m\in C$ for all $u\in L$, where $s,t,m$ are fixed positive integers. If $\gamma \ne 0$ or $d\ne 0$, then $a=0$ unless $R\subseteq M_2(F)$, the $2\times 2$ matrix ring over a field $F$. This result gives an affirmative answer to the open conjecture recently raised by Huang in [Derivations with annihilator conditions on Lie ideals in prime rings, J. Algebra Appl. 19 (2020) 2050025].

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, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]^t – [x, y]^m [α([x, y]),[x, y]]ⁿ [x, y]^s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s₄, the standard identity of degree 4. We also extend the results to semiprime rings.

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper.

Let $R$ be a $2$-torsion free prime ring, and $U$ a square closed Lie ideal of $R.$ Further let $G,$ and $F$ be generalized derivations associated with derivations $g$ and $d$, respectively on $R.$ If one of the following conditions holds: (i) $G(uv)+d(u)F(v)+uv\in Z(R),$ (ii) $G(uv)+d(u)F(v)+vu\in Z(R),$ (iii) $G(uv)+d(u)F(v)+uv\in Z(R),$ (iv) $G(vu)+F(u)F(v)+uv\in Z(R),$ (v) $G(uv)+d(v)F(u)+uv\in Z(R),$ for all $u,v\in U,$ then it is proved that either $g=d=0$ or $U\subseteq R.$

In this paper, we consider the notion of Jordan ideals and the Lie ideals in inverse semirings. It includes generalization of some well-known results of Herstein, in the setting of inverse semirings.

Let $R$ be a ring of characteristic different from $2$, $m,n,s\ge 1$ fixed positive integers, $L$ a noncentral Lie ideal of $R$ and $F:R\to R$ a nonzero generalized skew derivation of $R$.

We prove the following results:

• ^(a)If $R$ is prime and there exists $0\ne a\in R$ such that
  $$a{(F{(x)}^{m}F{(y)}^n-y^n{x}^m)}^s=0\forall x,y\in L$$
  then $R\subseteq M_2(K)$, the $2\times 2$ matrix ring over a field $K$.
• ^(b)If $R$ is semiprime and
  $${(F{(x)}^{m}F{(y)}^n-y^n{x}^m)}^s=0\forall x,y\in L$$
  then either $L$ centralizes a nonzero ideal of $R$ or $[s_4(x_1,\dots ,x_4),x_5]$ is a polynomial identity for $R$.

The aim of this paper is to introduce a new class of generalized centralizers on $C^{\ast }$-algebras which is termed as extended $(k_1,k_2)$-centralizer. We study different properties of this class of centralizers. Moreover, we define a class of $C^{\ast }$-algebras which contains the prime $C^{\ast }$-algebras as a subclass. Different properties of extended centralizers with respect to this class of $C^{\ast }$-algebras are discussed with suitable examples.