Prime Rings Research Articles

A generalization on derivations of Lie algebras

<p style='text-indent:20px;'>We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.</p>

M-potent commutators of skew derivations on Lie ideals

Abstract The purpose of this paper is to investigate the behavior of prime rings involving skew derivations with m-potent commutators on Lie ideals. In addition, we provide an example that shows that we cannot expect the same conclusion in case of semiprime rings. Also, we prove some other related results and present some open problems.

Generalized semiderivations in prime rings with algebraic identities

Let R be a prime ring with center Z(R). Suppose that R admits a generalized semiderivation F with non-zero associated derivation d. We investigate the commutativity of a prime ring R satisfying certain identities.

On “On derivations and commutativity of prime rings with involution”

Abstract In this note, we indicate some errors in [S. Ali, N. A. Dar and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J. 23 2016, 1, 9–14] and present the correct versions of the erroneous results.

Linear and multilinear functional identities in a prime ring with applications

This paper approaches some universal-algebraic properties of the two kinds of multilinear functions [Formula: see text] and [Formula: see text] in a prime ring [Formula: see text], where [Formula: see text] are variable elements, [Formula: see text]. We shall demonstrate an algebraic procedure of deriving necessary and sufficient conditions for the two multilinear functional identities [Formula: see text] and [Formula: see text] to hold for all [Formula: see text], [Formula: see text]. Subsequently, we use these multilinear functional identities to describe the invariance properties of the products [Formula: see text] [Formula: see text], [Formula: see text], [Formula: see text] with respect to the eight commonly-used types of generalized inverses of two MP-invertible elements [Formula: see text] and [Formula: see text] in a prime ring [Formula: see text] with an identity element 1 and ∗-involution.

Jordan Triple Higher (σ,τ)-Homomorphisms on Prime Rings

In this paper, the concept of Jordan triple higher -homomorphisms on prime
 rings is introduced. A result of Herstein is extended on this concept from the ring into the prime ring . We prove that every Jordan triple higher -homomorphism of ring into prime ring is either triple higher -homomorphism or triple higher -anti-homomorphism of into .

On the structure of generalized Jordan *-derivations of prime rings

Let be a noncommutative prime ring with involution and let be the maximal symmetric ring of quotients of In the present paper, we describe the structure of generalized Jordan *-derivations, i.e., additive mappings satisfying for all where d is an associated Jordan *-derivation of Precisely, we prove that under certain conditions any generalized Jordan *-derivation of is of the form for all where As a consequence, we show that any generalized Jordan *-derivation on a prime ring with involution is generalized X-inner, provided is not a PI-ring. Finally, we conclude our paper with a direction for further research.

On A Subring Of Prime Ring With Derivation


 
 
 Let R be a noncommutative prime ring of characteristic not 2, and let d be a nonzero derivation of R. We prove that the subring V of R generated by all [d(x), y], x,y \in R contains a nonzero two-sided ideal of R. 
 
 

Commutators and anti-commutators having automorphisms on Lie ideals in prime rings

In this manuscript, we discuss the relationship between prime rings and automorphisms satisfying differential identities involving commutators and anti-commutators on Lie ideals. In addition, we provide an example which shows that we cannot expect the same conclusion in case of semiprime rings.

On derivations involving prime ideals and commutativity in rings

The principal aim of this paper is to study the structure of quotient rings R/P where R is an arbitrary ring and P is a prime ideal of R. Especially, we will establish a relationship between the structure of this class of rings and the behaviour of derivations satisfying algebraic identities involving prime ideals. Some well-known results characterizing commutativity of (semi)-prime rings have been generalized.

An Engel condition with two generalized derivations in prime rings

Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that for all where are fixed positive integers. Then, one of the following conditions holds: there exist such that and for all m = s, n = t and there exist and such that g(x) = xa and for all C is a finite field, the matrix ring over C for some integer and there exist such that g(x) = xa and h(x) = ux for all

Generalized Jordan Triple -Higher Homomorphisms on Prime Rings

Herstein proved that any Jordan homomorphism onto a prime ring of characteristic of different from 2 and 3 is either a homomorphism or an anti-homomorphism. In this paper the concept of Generalized Jordan triple ( )–Higher Homomorphisms (GJT( )-HH) where and are two commuting homomorphisms are introduced as follows: A family of additive mappings of into is said to be a Generalized Triple -Higher Homomorphism (GT( )-HH) if there exist a triple -higher homomorphism (T ( ) – HH) such that for each and for all , we have: and is said to be the relating triple -HH. We will primarily extend the result of Herstein on it. It should be proved that every GJT()-HH of ring into prime ring is either GT( )-HH or triple ( ) higher anti-homomorphism (T( )-HAH).

Special Mappings with Central Values on Prime Rings

In this paper we investigate commutativity of prime rings with involution ∗ of the second kind in which endomorphisms satisfy certain algebraic identities. Furthermore, we provide examples to show that the various restrictions imposed by the hypotheses of our theorems are not superfluous.

Generalised Inner Derivations in Semi Prime Rings

Let A be any ring and f(xy) = f(x)y+xha(y), where f be any generalised inner derivation(G.I.D ) a be the fixed element of A. In this paper, it is shown that (i) ha must necessarily be a derivation for semi prime ring A. (ii) ∃ no generalized inner derivations f : A → A such that f(x ◦ y) = x ◦ y or f(x ◦ y) + x ◦ y = 0 ∀ x,y ∈ A, We have proved Havala [2] def. p.1147, Herstein [3] Lemma 3.1 p. 1106 as corollaries, along with other results.

On Posner’s second theorem for semirings with involution

EC Posner [14] proved a commutativity result known as Posner’s second theorem for a derivation centralizing on a prime ring. In this paper we prove some results for *-Jordan ideals of MA-semirings with involution. Main objective of this paper is to prove the aforesaid result for *-prime MA-semirings.

Generalized Derivations and Generalization of Co-commuting Maps in Prime Rings

Suppose that $R$ is a prime ring of characteristic different from $2$ with Utumi quotient ring $U$, $C = Z(U)$ the extended centroid of $R$, and $f(x_1,\ldots,x_n)$ a noncentral multilinear polynomial over $C$. If $F$, $G$ and $H$ are three nonzero generalized derivations of $R$ such that \[ F\big( G(f(X)) f(X) \big) = f(X) H(f(X)) \] for all $X = (x_{1},\ldots,x_{n}) \in R^n$, then we describe the nature of the maps $F$, $G$ and $H$.

Derivations with annihilator conditions on multilinear polynomials

Let R be a prime algebra over a commutative ring K, let I be a nonzero ideal of R and let be a multilinear polynomial over K in k non-commuting indeterminates which is not central-valued on R. Suppose that and d is a derivation of R such that for all , where , s, t, m are fixed positive integers and is the centre of R. It is shown that a = 0 or d = 0 except when , the matrix ring over a field F. This result gives a natural generalization of several well-known theorems in the literature. Moreover, with this we give 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. 2020;19:2050025].

Annihilators of power values of b-generalized derivations in prime rings

Let $R$ be a prime ring with extended centroid $C$ and maximal left ring of quotients $Q_{ml}(R)$. For a nonzero element $b\in R$ let $F:R\rightarrow R$ be a right generalized $b$-derivation associated with the map $d$ of $R$. Suppose that $s\left(F(x)\right)^n=0$ for all $x\in R$ where $s$ is a nonzero element in $R$ and $n\geq 1$ is a fixed positive integer. Then there exist some $c\in Q{ml}(R)$ and $\beta \in C$ such that $d(x)=ad_c(x)$, $F(x)=(b+\beta)xb$ for all $x\in R$ and either $s(c+\beta)=0$ or $b(c+\beta)=0$.

Representing nilpotent matrices as single commutators

We will show that every nilpotent element N in , the ring of square matrices over a division ring, can be presented as a single commutator, that is, for some matrices A, B in . We will also construct an example illustrating that there exists a prime ring with unity over which some nilpotent matrices cannot be presented as commutators.

On graded primitive Leavitt path algebras

Graded primitive Leavitt path algebras of arbitrary directed graphs over a field [Formula: see text] are completely characterized by means of graphical conditions. Necessary and sufficient conditions are given under which a graded prime Leavitt path algebra becomes graded primitive and this leads to answering the graded version of a question of Kaplansky on von Neumann regular prime rings in the context of Leavitt path algebras.