Semiprime Rings Research Articles

Strong inner and strong reflexive inverses in semiprime rings

Motivated by some recent work on von Neumann regular elements in semiprime rings, we study how strongly regular elements of semiprime rings are related in terms of their sets of strong inner inverses and strong reflexive inverses.

Characterization of additive mappings on semiprime rings

Characterization of additive mappings on semiprime rings

Generalized derivations with nilpotent values on Lie ideals in semiprime rings

Generalized derivations with nilpotent values on Lie ideals in semiprime rings

The Reflexive Condition on Skew Monoid Rings

This paper is devoted to introducing and studying two concepts, σ-skew strongly M-reflexive and σ-skew strongly M-nil-reflexive on monoid rings, which are generalizations of strongly M-reflexive and M-compatible. The paper covers the basic properties of skew monoid rings of the form R∗M. It is shown that if R is a left AP P (quasi Armendariz, semiprime rings, respectively), then R is σ-skew strongly M-reflexive. Moreover, if R is a NI-ring and M is a u.p-monoid, then R is σ-skew strongly M-nil-reflexive. Additionally, under some necessary and sufficient conditions, a skew monoid ring R ∗ M is proven to be σ-skew strongly M-nil-reflexive when σ : M → Aut(R) is a monoid homomorphism. Furthermore, if R is a left AP P, then the upper triangular matrix ring Tn(R) is σ ̄-skew strongly M-nil-reflexive, where n is a positive integer. Finally, the paper provides some examples and discusses related results from the subject.

RELATIONS OF MULTIPLICATIVE GENERALIZED (α,β)- REVERSE DERIVATION AND α- COMMUTING MAPS

In this paper, properties of ideal I of semiprime ring R with multiplicative generalized (α,β)- reverse derivation with determined not necessarily additive map d is studied. We generalized previous studies for different derivations to multiplicative generalized (α,β)- reverse derivation F. We show that [β(p),d(p)]I=0 for all p∈I or [d(p)],α(p)]I=0 for all p∈I under the given different conditions. Also, we give the relationship between map d and anti-automorphism α of semiprime ring R and automorphism β of semiprime ring R. Under the given different conditions, we examine whether d is α- commuting on ideal I or β- commuting on ideal I and obtain new results.

On skew-commuting generalized skew derivations in prime and semiprime rings

Let [Formula: see text] be a noncommutative prime ring of characteristic different from [Formula: see text], with right Martindale quotient ring [Formula: see text] and extended centroid [Formula: see text]. Let [Formula: see text] be fixed integers and [Formula: see text] a nonzero generalized skew derivation of [Formula: see text]. In this paper, we investigate the set [Formula: see text] and prove that its left annihilator in [Formula: see text] is identically zero. Using the above result, we also study the identity [Formula: see text] for semiprime rings.

Multiplicative (generalized)-reverse derivations in rings and Banach algebras

Abstract In this work, the subject of ideal in a semiprime ring with multiplicative (generalized)- reverse derivations studied is included. We give new essential results for researchers in this field and generalize some results found in the literature. Also, the application of continuous reverse derivations in Banach algebras is discussed for the first time.

Commutativity theorems on Lie ideals with symmetric bi-derivations in semiprime rings

Let [Formula: see text] be a semiprime ring, [Formula: see text] a square-closed Lie ideal of [Formula: see text] and [Formula: see text] a symmetric bi-derivation and [Formula: see text] be the trace of [Formula: see text] In this paper, we shall prove that [Formula: see text] contains a nonzero central ideal if any one of the following holds: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] (v) [Formula: see text] (vi) [Formula: see text] (vii) [Formula: see text] (viii) [Formula: see text] (ix) [Formula: see text] (x) [Formula: see text] (xi) [Formula: see text] and (xii) [Formula: see text], for all [Formula: see text]

A Characterization of Semiprime Rings with Homoderivations

This paper is focused on the commutativity of the laws of semiprime rings, which satisfy some algebraic identities involving homoderivations on ideals. It provides new and notable results that will interest researchers in this field, such as “R contains a nonzero central ideal if R admits a nonzero homoderivation δ on I such that δ(I)⊆Z where R is a semiprime ring with center Z and I a nonzero ideal of R”. Moreover, the research also generalizes some results previously published in the literature, including derivation on prime rings using homoderivation semiprime rings. It also demonstrates the necessity of hypotheses operationalized in theorems by an example. Finally, the paper discusses how the results herein can be further developed in future research.

ON NIL SKEW GENERALIZED POWER SERIES REFLEXIVE RINGS

Let $R$ be a ring and $(S, \leq)$ a strictly ordered monoid. In this paper, we deal with a new approaches to reflexive property for rings by using nilpotent elements. In this direction we introduce the notions of $(S, \omega)$-reflexive and $(S, \omega)$-$nil$-reflexive. Examples are given that, $(S, \omega)$-$nil$-reflexive is not $(S, \omega)$-reflexive. Under some suitable conditions, we proved that, if $R$ is a right $APP$-ring, then $R$ is $(S, \omega)$-reflexive and $R$ be a semiprime ring with the $ACC$ on left annihilator ideals, $(S, \leq)$ an $a.n.u.p.$-monoid, then $R$ is $(S, \omega)$-reflexive. Also, we proved that, $R$ is $(S, \omega)$-$nil$-reflexive if and only if $R/I$ is $(S, \overline{\omega})$-$nil$-reflexive, $R$ is $(S, \omega)$-$nil$-reflexive if and only if $T_{n}(R)$ is $(S, \omega)$-$nil$-reflexive and we will show that, if $R$ is a right Noetherian ring, then $R$ is $(S, \omega)$-$nil$-reflexive. Moreover, we investigate ring extensions which have roles in ring theory.

Identities on additive mappings in semiprime rings

Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : R\to R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $\delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m \delta(x^n)$$ for each $x$ in $R$ and $k\in \{2, m, n, (n+m-1)!\}$ and at last an application on Banach algebra is presented.

Jordan *-derivations on standard operator algebras

LetH be a real or complex Hilbert space with dim(H) > 1, B(H) be algebra of all bounded linear operators on H and A(H) ? B(H) be a standard operator algebra on H. If D : A(H) ? B(H) is a linear mapping satisfying D(An+1) = Pn i=0 AiD(A)(A*)n?i for all A ? A(H), then D is a Jordan *-derivation on A(H). Later, we discuss some algebraic identities on semiprime rings.

On 2-Commutative Derivations of Semi-prime Rings

The main purpose of this paper is to study some results concerning the generalized derivation D defined on semi-prime ring R, we obtain a derivation d which commuting and 2-commuting on R. Also, we present many examples to clarify the validity of our work.

Additive mappings satisfying algebraic identities in semiprime rings

Additive mappings satisfying algebraic identities in semiprime rings

CoSemiprime Ring

In this paper ,we introduce a new type of rings,namely cosemiprime ringas a generalization of semiprime ring,where a ring C is said to be cosemiprime ring if I= for each ideal I of C .Some properties of this type of rings are obtained . Also,we determin the relationship among types of ideals in this ring.

Semiprime rings with generalized homoderivations

This study develops some results involving generalized homoderivation in semiprime rings and investigates the commutativity of semiprime rings admitting generalized homoderivations of ring R satisfying certain identities and some related results have also been discussed.

Some identities in quotient rings

Let R be an associative ring, P a prime ideal of R: In this paper, we study the structure of the ring R=P and describe the possible forms of the generalized derivations satisfying certain algebraic identities on R: As a consequence of our theorems, we first investigate strong commutativity preserving generalized derivations of prime rings, and then examine the generalized derivations acting as (anti)homomorphisms in prime rings. Some commutativity theorems also given in semi-prime rings.

On multiplicative centrally-extended maps on semi-prime rings

ABSTRACT In this paper, we show that for semi-prime rings of two-torsion free and 6-centrally torsion free, given a multiplicative centrally-extended derivation δ and a multiplicative centrally-extended epimorphism ϕ we can find a central ideal K and maps such as Δ and Φ are multiplicative derivation and multiplicative epimorphism, respectively.

SYMMETRIC GENERALIZED REVERSE (α, 1)-BIDERIVATIONS IN RINGS

Let R be a ring and α be an endomorphism of R. Then, we introduce the notions of generalized reverse (α, 1)-derivation and that of symmetric generalized reverse (α, 1)-biderivation. It is shown that if a semiprime ring admits a generalized reverse (α, 1)-derivation with an associated reverse (α, 1)-derivation d, then d maps R into Z(R) and also that if a non-commutative prime ring admits a generalized reverse (α, 1)-derivation F with an associated reverse (α, 1)-derivation d, then F is reverse left α-multiplier on R. Analogous results have been proved for a symmetric generalized reverse (α, 1)-biderivation.

Some identities involving generalized (α,β)-derivations in prime and semiprime rings

Let [Formula: see text] be a prime or semiprime ring, [Formula: see text] a nonzero left-sided ideal of [Formula: see text] and [Formula: see text] be two automorphisms of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two generalized [Formula: see text]-derivations of [Formula: see text] associated with [Formula: see text]-derivations [Formula: see text] and [Formula: see text], respectively. The purpose of this paper is to investigate the following identities: (1) [Formula: see text]; (2) [Formula: see text]; (3) [Formula: see text]; (4) [Formula: see text]; (5) [Formula: see text]; (6) [Formula: see text]; (7) [Formula: see text]; (8) [Formula: see text]; (9) [Formula: see text]; (10) [Formula: see text]; (11) [Formula: see text]; (12) [Formula: see text], for all [Formula: see text].