Semiprime Rings Research Articles

Symmetric Reverse n-Derivations on Ideals of Semiprime Rings

This paper focuses on examining a new type of n-additive map called the symmetric reverse n-derivation. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse n-derivations. This study aims to deepen our understanding of these symmetric maps and their properties within mathematical structures.

D-Semiprime Rings

Let R be an associative and 2-torsion-free ring with an identity. in this work, we will generalize the results of differentially prime rings in [18] by applying the hypotheses in a differentially semiprime rings. In particular, we have proved that if R is a D-semiprime ring, then either R is a commutative ring or D is a semiprime ring.

Structure of zero-divisors to go up to related ideals

There are many ways for zero-dividing polynomials to go up to zero-dividing ideals when the base rings are IFP. The importance of these in ring theory leads us to consider the following ring conditions and study new useful roles of matrices for ring theory. Let R be a ring and a , b ∈ R \\ { 0 } . The first is the condition (*) that if ab = 0 then Ib = 0 for some nonzero ideal I ⊆ RaR of R or aJ = 0 for some nonzero ideal J ⊆ RbR of R. It is shown that from given any IFP ring, there can be constructed a non-IFP ring with the condition (*). We prove that a semiprime ring R with the condition (*) is both right and left nonsingular. The second is the condition (**) that if ab = 0 then IJ = 0 for some nonzero ideals I ⊆ RaR and J ⊆ RbR of R. We prove that every ring can be a subring of rings with the condition (**), that if R is an irreducible ring with the condition (**) then R is either a domain or non-semiprime, and that the condition (**) passes to polynomial rings when the base ring is semiprime. Various sorts of examples are given to illustrate and delimit the results obtained.

Chain conditions on non-parallel submodules

In this paper, we investigate modules with ascending and descending chain conditions on non-parallel submodules. We call these modules np-Noetherian and np-Artinian respectively, and give structure theorems for them. It is proved that any np-Artinian module is either atomic or finitely embedded. Also, we give a sufficient condition for np-Noetherian (resp., np-Artinian) modules to be Noetherian (resp., Artinian). We study ascending (resp., descending) chain condition up to isomorphism on non-parallel submodules as npi-Noetherian (resp., npi-Artinian) modules and characterize these modules. It is shown that any npi-Noetherian module has finite type dimension. Next, we investigate some properties of semiprime right np-Artinian (resp., npi-Artinian) rings. In particular, it is proved that if $ R $ semiprime ring such that $ J(R) $ is not atomic, then $ R $ is right np-Artinian if and only if it is semisimple. Further, it is shown that if $ R $ is a semiprime right npi-Artinian ring, then either $ Z(R) $ is atomic or $ R $ is right non-singular. Finally, we investigate when np-Artinian (resp., np-Noetherian) rings and ne-Artinian (resp., ne-Noetherian) rings coincide.

Center-like Subsets in Semiprime Rings with Multiplicative Derivations

We introduce center-like subsets Z∘*(A,d),Z∘**(A,d), where A is the ring and d is the multiplicative derivation. In the following, we take a new derivation for the center-like subsets existing in the literature and establish the relations between these sets. In addition to these new sets, the theorems are generalized as multiplicative derivations instead of the derivations found in previous studies. Additionally, different proofs are provided for different center-like sets. Finally, we enrich this article with examples demonstrating that the hypotheses we use are necessary.

Zero-Power Valued Mappings with Dependent Element of Associative Rings

Let R` be a semi prime rings. An additive mapping L: R`→R` is called a zero-power valued on V over L is preserving V and if for each p belongs to V, that’s meaning, there exists a positive integer n(p) > 1 such that L^(n(p))(p) = 0. The main purpose of this article is to study the notion of an additive mapping which is named zero-power valued mapping with dependent element of a semi prime ring. As a factual information, we deliver some results concerning that.

Action of higher derivations on semiprime rings

Abstract Let m , n {m,n} be the fixed positive integers and let ℛ {\mathcal{R}} be a ring. In 1978, Herstein proved that a 2-torsion free prime ring ℛ {\mathcal{R}} is commutative if there is a nonzero derivation d of R such that [ d ⁢ ( ϱ ) , d ⁢ ( ξ ) ] = 0 {[d(\varrho),d(\xi)]=0} for all ϱ , ξ ∈ R {\varrho,\xi\in R} . In this article, we study the above mentioned classical result for higher derivations and describe the structure of semiprime rings by using the invariance property of prime ideals under higher derivations. Precisely, apart from proving some other important results, we prove the following. Let ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} and ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} be two higher derivations of semiprime ring ℛ {\mathcal{R}} such that [ d n ⁢ ( ϱ ) , g m ⁢ ( ξ ) ] ∈ Z ⁢ ( ℛ ) {[d_{n}(\varrho),g_{m}(\xi)]\in Z(\mathcal{R})} for all ϱ , ξ ∈ ℐ {\varrho,\xi\in\mathcal{I}} , where ℐ {\mathcal{I}} is an ideal of ℛ {\mathcal{R}} . Then either ℛ {\mathcal{R}} is commutative or some linear combination of ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero or some linear combination of ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero. We enrich our results with examples that show the necessity of their assumptions. Finally, we conclude our paper with a direction for further research.

Orthogonal Generalized Symmetric Reverse Bi-(????, τ)-Derivations of Semi Prime Ring

Orthogonal Generalized Symmetric Reverse Bi-(????, τ)-Derivations of Semi Prime Ring

Multiplicative (generalized)-derivations acting on left sided ideals with annihilator conditions in semiprime rings

Multiplicative (generalized)-derivations acting on left sided ideals with annihilator conditions in semiprime rings

On the structure of prime and semiprime rings with generalized skew derivations

Let [Formula: see text] be a ring of characteristic different from [Formula: see text], [Formula: see text] fixed positive integers, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] a nonzero generalized skew derivation of [Formula: see text]. We prove the following results: (a) If [Formula: see text] is prime and there exists [Formula: see text] such that [Formula: see text] then [Formula: see text], the [Formula: see text] matrix ring over a field [Formula: see text]. (b) If [Formula: see text] is semiprime and [Formula: see text] then either [Formula: see text] centralizes a nonzero ideal of [Formula: see text] or [Formula: see text] is a polynomial identity for [Formula: see text].

Generalized Jordan triple derivations associated with Hochschild 2-cocycles in semiprime rings

Generalized Jordan triple derivations associated with Hochschild 2-cocycles in semiprime rings

A Note on the Trace of Generalized Permuting Tri-Derivations

Many researchers have studied permuting tri-derivation and generalized derivation in prime or semi-prime rings, BCK-algebras, lattices, d-algebras, MV-algebras and many algebraic structures. Later, they introduced the concept of generalized permuting tri-derivation by combining the concepts of generalized derivation and permuting tri-derivation. In this article, we have examined the properties of generalized permuting triderivation by adding conditions on their traces in prime or semi-prime rings. In addition, we examined the properties of two permuting triderivation by giving a relation between their traces.

A New Study on Generalized Reverse Derivations of Semi-prime Ring

The aim of this paper is to extend the ideas from Generalized reverse derivation to Generalized (α, β)-reverse derivations on Semi-prime ring. We prove that, if 0≠d be reverse derivation in R and a Generalized (α, β)-reverse derivation g, then g is β-strong commutative preserved. Next we can prove that R is commutative.

Results on prime and semi-prime rings with skew and generalized reverse derivations

For this study, R represents a semiprime ring or a prime ring, as the case may be. The ring R is said to be semiprime if for any a ∈ R, aRa = {0}, implies a = 0. R is a prime ring if aRb = {0}, implies a = 0 or b = 0, ∀ a, b ∈ R. By assuming that d is a skew derivation with an automorphism β: R→R associated with it, we prove some results on skew-derivations for semi-prime rings. In particular, we show that for a skew-derivation, if d(a)d(b) ± ab = 0, ∀ a, b ∈ R then d = 0. Also, by introducing new differential identities, we establish that a prime ring with a generalized reverse derivation defined on it is commutative.

A description of linear mappings in semiprime rings with involution

The main purpose of this paper is to descriptive the action of the linear mappings in semi-prime rings and prime ring with involution. More precisely, we establish some results for centralizer mappings (resp. commuting mappings) of a semi-prime *- ring and *-prime ring ℛ.

IDENTITIES WITH MULTIPLICATIVE GENERALIZED (α,α)-DERIVATIONS OF SEMIPRIME RINGS

Let R be a semiprime ring and α be an automorphism of R. A mapping F : R → R (not necessarily additive) is called multiplicative generalized (α,α)-derivation if there exists a unique (α,α)-derivation d of R such that F(xy) = F(x)α(y) + α(x)d(y) for all x,y ∈ R. In the present paper, we intend to study several algebraic identities involving multiplicative generalized (α,α)-derivations on appropriate subsets of semiprime rings and collect the information about the commutative structure of these rings.

One-sided nilpotent semicommutative properties of rings

In this paper, we study the semicommutative properties of rings with one-sided nilpotent structures. The concepts of left and right nilpotent semicommutative rings are defined and studied, which are a generalization of semicommutative rings. We show that the class of one-sided nilpotent semicommutative rings is strictly placed between the class of semicommutative rings and that of nil-semicommutative rings. As applications, we give some characterizations of semiprime rings from the point of view of left nilpotent semicommutative rings.

Symmetric n-Additive Mappings Admitting Semiprime Ring

Symmetric n-Additive Mappings Admitting Semiprime Ring

Generalized derivations with nilpotent values in semiprime rings

Let R be a (semi-) prime ring with extended centroid C, let f(X 1, … , Xk ) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x 1, … , xk )) m − γf(x 1, … , xk ) n ) s = 0 for all x 1, … , xk ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.]

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.