Semiprime Rings Research Articles

Reversible ring property via idempotent elements

Abstract Regarding the question of how idempotent elements affect the reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce right (resp., left) 𝑒-reversible rings. We show that this concept is not left-right symmetric. Basic properties of right 𝑒-reversibility in a ring are provided. Among others, it is proved that if 𝑅 is a semiprime ring, then 𝑅 is right 𝑒-reversible if and only if it is right 𝑒-reduced if and only if it is 𝑒-symmetric if and only if it is right 𝑒-semicommutative. Also, for a right 𝑒-reversible ring 𝑅, 𝑅 is a prime ring if and only if it is a domain. It is shown that the class of right 𝑒-reversible rings lies strictly between that of 𝑒-symmetric rings and right 𝑒-semicommutative rings. Some extensions of rings are studied in terms of 𝑒-reversibility.

Generalized Commuting Mapping in Prime and Semiprime Rings

Let R be an associative ring. The essential purpose of the present paper is to introduce the concept of generalized commuting mapping of R. Let U be a non-empty subset of R, a mapping : R R is called a generalized commuting mapping on U if there exist a mapping :R R such that =0, holds for all U. Some results concerning the new concept are presented.

On annihilator ideals in skew power series rings

The purpose of this paper is to study the behavior of the IN and SA property with respect to the skew power series and Laurent series extensions. For an α-compatible ring R it is shown that the skew power series ring is right SA if and only if R is right SA and satisfies the (SQA2) condition. Also it is shown that a semiprime ring R is right SA if and only if is right SA. Moreover, if α is an automorphism, then we have similar results for skew Laurent series ring Furthermore, if R is a reduced left Noetherian ring, then R is left IN if and only if is left IN if and only if is left IN. Also we present some examples to show that if we relax each assumption on R, our results can fail.

Note on Lie ideals with symmetric bi-derivations in semiprime rings

Note on Lie ideals with symmetric bi-derivations in semiprime rings

On the Multiplicative (Generalised) (α, α)−Derivations of Semiprime Rings

The algebraic properties and identities of a semiprime ring are investigated with the help of the multiplicative (generalised)-(α, α)-reverse derivation on the non-empty ideal of the semiprime ring.

Structure of a quotient ring R/P and its relation with generalized derivations of R

The fundamental aim of this paper is to investigate the structure of a quotient ring R/P where R is an arbitrary ring and P is a prime ideal of R. More precisely, we will characterize the commutativity of R/P via the behavior of generalized derivations of R satisfying algebraic identities involving the prime ideal P. Moreover, various wellknown results characterizing the commutativity of prime (semi-prime) rings have been extended. Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.

Some identities related to multiplicative (generalized)-derivations in prime and semiprime rings

Some identities related to multiplicative (generalized)-derivations in prime and semiprime rings

Orthogonal Semiderivations and Symmetric Bi-semiderivations in Semiprime Rings

In this paper, orthogonality for symmetric bi-semiderivations is defined and some results are obtained when two symmetric bi-semiderivations are orthogonal. Also, this paper gives the notion of orthogonality between semiderivations and symmetric bi-semiderivations of a 2-torsion free semiprime ring and offers some results of orthogonality.In this paper, orthogonality for symmetric bi-semiderivations is defined and some results are obtained when two symmetric bi-semiderivations are orthogonal. Also, this paper gives the notion of orthogonality between semiderivations and symmetric bi-semiderivations of a 2-torsion free semiprime ring and offers some results of orthogonality.

Some Identities Involving Multiplicative (Generalized)(α; 1)-Derivations in Semiprime Rings

Let R be a semiprime ring, I a non-zero ideal of R and α be an automorphism of R. A map F : R to R is said to be a multiplicative (generalized)(α, 1)-derivation associated with a map d : R to R such that F (xy) = F (x) α (y) + xd (y), for all x, y in R. In the present paper, we shall prove that R contains a non-zero central ideal if any one of the following holds: (i) F [x, y] ± [x,y] =0; (ii) F (xoy)±α(xoy) = 0; (iii) F [x, y] = [F (x) , y]α;1 ; (iv) F [x, y] =(F (x) oy)α;1 ; (v) F (xoy) = [F(x) , y]α;1 and (vi)F (xoy) = (F (x) oy)α;1, for all x, y in I.

Characterization of higher commutators

Let R be a ring having the property that every proper ideal of R is contained in a maximal ideal of R (in particular, if R is finitely generated as an ideal). Generalizing several known results, we characterize higher commutators V of R whenever R is generated by V (respectively, [R,V]) as an ideal. In particular, if V is a higher commutator of a unital ring R with 1∈V, then V is equal to either R or [R,R], or [[R,R],[R,R]]. Given a semiprime ring R, which is generated by [R,R], all higher commutators of R are obtained if R possesses a central higher commutator. We also characterize all higher commutators of Mn(D) for n≥2 when D is a unital commutative ring. In addition, if D is 2-torsion free and 2D⊊D, then M2(D) has infinitely many higher commutators.

Generalized multiplicative α-skew derivations on rings

Let R be a semiprime (or prime) ring and U be a nonzero ideal of R. In the present paper, we study the notions of multiplicative generalized α-skew derivations on ideals of R and prove that if R admits a multiplicative generalized α-skew derivation G associated with a nonzero additive map d and an automorphism α, then d is necessarily an α-skew derivation of R. Also, we study the structure of a semiprime ring admitting a multiplicative generalized α-skew derivation satisfying more specific algebraic identities. Moreover, we also provide examples to show that the assumed restrictions cannot be relaxed

The socle of Leavitt path algebras over a semiprime ring

The Reduction Theorem in Leavitt path algebra over a commutative unital ring is very important to prove that the Leavitt path algebra is semiprime if and only if the ring is also semiprime. Any minimal ideal in the semiprime ring and line point will construct a left minimal ideal in the Leavitt path algebra. Vice versa, any left minimal ideal in the semiprime Leavitt path algebra can be found both minimal ideal in the semiprime ring and line point that generate it. The socle of semiprime Leavitt path algebra is constructed by minimal ideals of the semiprime ring and the set of all line points.

Centralizing n‐Homoderivations of Semiprime Rings

We introduce the notion of n‐homoderivation on a ring ℜ and show that a semiprime ring ℜ must have a nontrivial central ideal if it admits an appropriate n‐homoderivation which is centralizing on some nontrivial one‐sided ideal. Under similar hypotheses, we prove commutativity in prime rings.

Centrally Extended α‐Homoderivations on Prime and Semiprime Rings

We present a new type of mappings called centrally extended α‐homoderivations of a ring ℜ (i.e., a map H from ℜ into ℜ which satisfies H(x + y) − H(x) − H(y) ∈ Z(ℜ) and H(xy) − H(x)H(y) − H(x)α(y) − α(x)H(y) ∈ Z(ℜ) for any x, y ∈ ℜ) where α is a mapping of ℜ and discuss the relationship between these mappings and other related mappings. Also, we study their effect on the center of a ring, and we give several outcomes on the commutativity of the ring.

Functional equations related to higher derivations in semiprime rings

Abstract We investigate the additivity and multiplicativity of centrally extended higher derivations and show that every centrally extended higher derivation of a semiprime ring with no nonzero central ideals is a higher derivation. Moreover, we study preservation of the center of the ring by a centrally extended higher derivation.

(ϕ, φ)-derivations on semiprime rings and Banach algebras

Abstract Let ℛ be a semiprime ring with unity e and ϕ, φ be automorphisms of ℛ. In this paper it is shown that if ℛ satisfies 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) 2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right) for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 is an (ϕ, φ)-derivation. Moreover, this result makes it possible to prove that if ℛ admits an additive mappings 𝒟, 𝒢 : ℛ → ℛ satisfying the relations 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) 2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right) 2 𝒢 ( x n ) = 𝒢 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) D ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) , 2\mathcal{G}\left( {{x^n}} \right) = \mathcal{G}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right), for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 and 𝒢 are (ϕ, φ)--derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.

NOTES ON LEFT IDEALS OF SEMIPRIME RINGS WITH MULTIPLICATIVE GENERALIZED (alpha,alpha)-DERIVATIONS

Let R be a 2-torsion free semiprime ring, I a nonzero left ideal of R, alpha an automorphism on R and F a multiplicative (generalized) (alpha,alpha)-derivation of R. This note we gave the description of commutativity of semiprime rings with help of some identities involving a multiplicative generalized (alpha,alpha)-derivation and a nonzero left ideal of R.

Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution

Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution

On Rings of Weak Global Dimension at Most One

A ring R is of weak global dimension at most one if all submodules of flat R-modules are flat. A ring R is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left ideals) of R is distributive. Jensen has proved earlier that a commutative ring R is a ring of weak global dimension at most one if and only if R is an arithmetical semiprime ring. A ring R is said to be centrally essential if either R is commutative or, for every noncentral element x∈R, there exist two nonzero central elements y,z∈R with xy=z. In Theorem 2 of our paper, we prove that a centrally essential ring R is of weak global dimension at most one if and only is R is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings.

On The Identity d(x) = λx + ζ(x)

The main purpose of this paper is to study and investigate some results concerning generalized derivation D on semiprime ring R, where d a derivation on R and R has a cancellation property with identity. Then under certain conditions we prove that there exist λ C and an additive mapping ζ: R →C such that d(x) = λx + ζ(x) for all x R. Mathematics Subject Classification: 16W25, 16N60, 16U80.