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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. [ABSTRACT FROM AUTHOR]

Let R be an associative and 2-torsion-free ring with an identity. in this work, we will generaliz 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The objective of this research paper is to justify that an additive mapping F from a semiprime ring R to itself will be ϕ-centralizer having a suitable torsion restriction on R if it satisfies certain algebraic equations. [ABSTRACT FROM AUTHOR]

In the present manuscript our intention is to prove the following: Suppose that R is a semiprime ring and n is a positive integer. If d , F : R → R are two additive mappings satisfying the identity F (r 2 n) = F (r n) r n + r n d (r n) , ∀ r ∈ R , then d will be a derivation and F will be a generalized derivation associated with d on R. Alongside, it is shown that F 1 is generalized left derivation with associated left derivation d 1 on R if they satisfy the algebraic identity F 1 (r 2 n) = r n F 1 (r n) + r n d 1 (r n) , ∀ r ∈ R . These theorems have been validated by providing instances demonstrating that they are not inconsequential. Moreover, we furnish an application in context with Banach algebra. [ABSTRACT FROM AUTHOR]

In this article, we prove the following result. Let n≥3n\ge 3 be some fixed integer and let RR be a prime ring with char(R)≠(n+1)!2n−2{\rm{char}}\left(R)\ne \left(n+1)\!{2}^{n-2}. Suppose there exists an additive mapping D:R→RD:R\to R satisfying the relation 2n−2D(xn)=∑i=0n−2n−2ixiD(x2)xn−2−i+(2n−2−1)(D(x)xn−1+xn−1D(x))+∑i=1n−2∑k=2i(2k−1−1)n−k−2i−k+∑k=2n−1−i(2k−1−1)n−k−2n−i−k−1xiD(x)xn−1−i\begin{array}{rcl}{2}^{n-2}D\left({x}^{n})& =& \left(\mathop{\displaystyle \sum }\limits_{i=0}^{n-2}\left(\genfrac{}{}{0.0pt}{}{n-2}{i}\right){x}^{i}D\left({x}^{2}){x}^{n-2-i}\right)+\left({2}^{n-2}-1)\left(D\left(x){x}^{n-1}+{x}^{n-1}D\left(x))\\ & & +\mathop{\displaystyle \sum }\limits_{i=1}^{n-2}\left(\mathop{\displaystyle \sum }\limits_{k=2}^{i}\left({2}^{k-1}-1)\left(\genfrac{}{}{0.0pt}{}{n-k-2}{i-k}\right)+\mathop{\displaystyle \sum }\limits_{k=2}^{n-1-i}\left({2}^{k-1}-1)\left(\genfrac{}{}{0.0pt}{}{n-k-2}{n-i-k-1}\right)\right){x}^{i}D\left(x){x}^{n-1-i}\end{array} for all x∈R.x\in R. In this case, DD is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a prime ring with char(R)≠2{\rm{char}}\left(R)\ne 2 is a derivation.

Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic identity $ 2\mathcal{H}(a^{p+q}) = \mathcal{H}(a^p) a^q+ a^p \mathcal{D}(a^q)+\mathcal{H}(a^q) a^p+ a^q \mathcal{D}(a^p) $ for all $ a\in \mathcal{A} $, then $ \mathcal{H} $ is a generalized derivation with $ \mathcal{D} $ as an associated derivation on $ \mathcal{A} $. In addition to that, it is also proved in this article that $ \mathcal{H}_1 $ is a generalized left derivation associated with a left derivation $ \delta $ on $ \mathcal{A} $ if they fulfilled the algebraic identity $ 2\mathcal{H}_1(a^{p+q}) = a^p \mathcal{H}_1(a^q)+ a^q \delta(a^p)+a^q \mathcal{H}_1(a^p)+ a^p \delta(a^q) $ for all $ a \in \mathcal{A} $. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.

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.

Let R be a 2−torsion free semiprime ring, α an automorphism and U a noncentral square-closed Lie ideal of R. An additive mapping T : R → R is called a left (resp. right) α−centralizer of R if T(xy) = T(x)α(y) (resp. T(xy) = α(x)T(y)) holds for all x, y ∈ R. In this paper, we proved the following result: i) If T (uvu) = α (u) T (v) α (u) for all u, v ∈ U, then T is a left α−centralizer, ii) If T(uvu) = T(u)α(vu) (resp. T(uvu) = α(uv)T(u)) for all u, v ∈ U, then T is a left (resp. right) α-centralizer on U. [ABSTRACT FROM AUTHOR]

Let A be a (p + q)!-torsion free semiprime ring. We proved that if H, D : A → A are two additive mappings fulfilling the algebraic identity 2H(ap+q) = H(ap)aq + apD(aq) + H(aq)ap + aqD(ap) for all a e A, then H is a generalized derivation with D as an associated derivation on A. In addition to that, it is also proved in this article that H1 is a generalized left derivation associated with a left derivation δ on A if they fulfilled the algebraic identity 2H1(ap+q) = apH1(aq) + aqδ(ap) + aqH1(ap) + apδ(aq) for all a ∈ A. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the one-sided generalized (α, β)-reverse derivation of a ring R. Let R be a semiprime ring, ϱ be a non-zero ideal of R, α be an epimorphism of ϱ, β be a homomorphism of ϱ (α be a homomorphism of ϱ, β be an epimorphism of ϱ) and γ: ϱ → R be a non-zero (α, β)-reverse derivation. We show that there exists F: ϱ → R, an l-generalized (α, β)-reverse derivation (an r-generalized (α, β)-reverse derivation) associated with γ iff F(ϱ), γ(ϱ) ⊂ CR(ϱ) and F is an r-generalized (β, α)-derivation (an l-generalized (β, α)-derivation) associated with (β, α)-derivation γ on ϱ. This theorem generalized the results of A. Aboubakr and S. Gonzalez proved in [1, Theorem 3.1 and Theorem 3.2 ]. [ABSTRACT FROM AUTHOR]

Our intention in this manuscript is to study the commutative structure of the semiprime rings R on appropriate subsets of it. We observe several algebraic identities in the presence of multiplicative generalized skew-derivation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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.

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. [ABSTRACT FROM AUTHOR]

Let R be either a ring with 1 or a semiprime ring not necessarily with 1 and let Nn(R) be the n×n strictly upper triangular matrix ring over R, where n > 3 is an integer. We completely characterize additive maps f: Nn(R) → Nn(R) satisfying [f (x),x] = 0 for all x ∈ Nn(R). Our theorem naturally generalizes a recent result obtained by Bounds [3] for strictly upper triangular matrix rings over a field of characteristic 0. [ABSTRACT FROM AUTHOR]

The present paper aims to investigate the containment of nonzero central ideal in a ring $ \mathcal{R} $ when the trace of symmetric $ n $-derivations satisfies some differential identities. Lastly, we prove that in a prime ring $ \mathcal{R} $ of suitable torsion restriction, if $ \mathfrak{D}, \mathcal{G} : \mathcal{R}^n \rightarrow \mathcal{R} $ are two nonzero symmetric $ n $-derivations such that $ \mathcal{f}(\vartheta)\vartheta +\vartheta\mathcal{g}(\vartheta) = 0 $ holds $ \forall \; \vartheta \in \mathcal{W} $, a nonzero left ideal of $ \mathcal{R} $ where $ \mathcal{f} $ and $ \mathcal{g} $ are the traces of $ \mathfrak{D} $ and $ \mathcal{G} $, respectively, then either $ \mathcal{R} $ is commutative or $ \mathcal{G} $ acts as a left $ n $-multiplier. Finally, we characterize symmetric $ n $-derivations in terms of left $ n $-multipliers.

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. [ABSTRACT FROM AUTHOR]

A well-known result of Posner's second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we established the ∗-version of Vukman's theorem. Precisely, we describe the structure of quotient ring A / L , where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the ∗-version of Vukman's theorem, we show that if a 2-torsion free semiprime A with involution admits a nonzero ∗-centralizing derivation, then A contains a nonzero central ideal. This result is in the spirit of the classical result due to Bell and Martindale (Theorem 3). As the applications, we extended and unified several classical theorems. Finally, we conclude our paper with a direction for further research. [ABSTRACT FROM AUTHOR]

The present paper aims to investigate the containment of nonzero central ideal in a ring R when the trace of symmetric n -derivations satisfies some differential identities. Lastly, we prove that in a prime ring R of suitable torsion restriction, if D , G : R n → R are two nonzero symmetric n -derivations such that f (ϑ) ϑ + ϑ g (ϑ) = 0 holds ∀ ϑ ∈ W , a nonzero left ideal of R where f and g are the traces of D and G , respectively, then either R is commutative or G acts as a left n -multiplier. Finally, we characterize symmetric n -derivations in terms of left n -multipliers. [ABSTRACT FROM AUTHOR]

Generalized Munn rings exist extensively in the theory of rings. The aim of this note is to answer when a generalized Munn ring is primitive (semiprimitive, semiprime and prime, respectively). Sufficient and necessary conditions are obtained for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively). Also, we obtain sufficient and necessary conditions for a Munn ring over principal ideal domains to be prime (semiprime, respectively). Our results can be regarded as the generalizations of the famous result in the theory of rings that for a ring RR, RR is primitive (semiprimitive and semiprime, respectively) if and only if so is Mn(R){M}_{n}\left(R). As applications of our results, we consider the primeness and the primitivity of generalized matrix rings and generalized path algebras. In particular, it is proved that a path algebra is a semiprime if and only if it is semiprimitive.

Let R be a semiprime ring, U a square-closed Lie ideal of R and D : R × R → R a symmetric bi-derivation and d be the trace of D. In the present paper, we prove that the R contains a nonzero central ideal if any one of the following holds: i) d x y ± x g (y) ∈ Z , ii) [ d (x) , y ] = ± [ x , g (y) ] , iii) d (x) ∘ y = ± x ∘ g (y) , iv) [ d (x) , y ] = ± x ∘ g (y) , v) d ([ x , y ]) = [ d (x) , y ] + [ d (y) , x ] , vi) d (x y) ± x y ∈ Z , vii) d (x y) ± y x ∈ Z , viii) d (x y) ± [ x , y ] ∈ Z , ix) d (x y) ± x ∘ y ∈ Z , x) g (x y) + d (x) d (y) ± x y ∈ Z , xi) g (x y) + d (x) d (y) ± y x ∈ Z , xii) g ([ x , y ]) + [ d (x) , d (y) ] ± [ x , y ] ∈ Z , xiii) g (x ∘ y) + d (x) ∘ d (y) ± x ∘ y ∈ Z , for all x , y ∈ U , where G : R × R → R is symmetric bi-derivation such that g is the trace of G. [ABSTRACT FROM AUTHOR]

The main aim of this work is to present some special results related to relatively commuting mappings. We prove that if F is a non trivial derivation on a 2-torsion free prime ring R, then for any o ∈ R, the property that [[F(o),o], F(o)] is a central force F to be commuting on R. [ABSTRACT FROM AUTHOR]

Ring theory is one of the influential branches of abstract algebra. In this field, many algebraic problems have been considered by mathematical researchers who are working in this field. However, some new concepts have been created and developed to present some algebraic structures with their properties. Rings with derivations have been studied fifty years ago, especially the relationships between the derivations and the structure of a ring. By using the notatin of derivation, many results have been obtained in the literature with different types of derivations. In this paper, the concept of the derivation theory of a ring has been considered. This study presented the definition of Ring theory is one of the influential branches of abstract algebra. In this field, many algebraic problems have been considered by mathematical researchers who are working in this field. However, some new concepts have been created and developed to present some algebraic structures with their properties. Rings with derivations have been studied fifty years ago, especially the relationships between the derivations and the structure of a ring. By using the notatin of derivation, many results have been obtained in the literature with different types of derivations. In this paper, the concept of the derivation theory of a ring has been considered. This study presented the definition of (θ1,θ2) derivation pair and Jordan (θ1,θ2)-derivation pair on an associative ring Γ, and the relation between them. Furthermore, we study the concept of prime rings under this notion by introducing some of its properties where θ1 and θ2 are two mappings of Γ into itself.

In this paper we prove the following result: Let R be a prime ring with c h a r (R) ≠ 2 , 3 , 5 and let T : R → R be an additive mapping satisfying the relation T (x 4) = x T (x 2) x for all x ∈ R . In this case T is of the form T (x) = λ x for all x ∈ R and some fixed element λ ∈ C , where C is the extended centroid of R. [ABSTRACT FROM AUTHOR]

Let R be a ring with involution '*'. Next, let N0 be the set of all nonnegative integers, and D = .... The main aim of this paper is to characterize Jordan triple (α, β)-higher *-derivation of semiprime rings with involution. As an application, we prove that every Jordan triple (α, β)-higher *-derivation onto a 6-torsion free semiprime ring is a Jordan higher *-derivation. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity e and let F;D and T be additive maps from R into itself. The main aim of this article is to obtain some identities involving at the most 4 terms satisfied by F;D and T on the semiprime ring R which is m-torsion free for some m so that F;D and T become a Jordan generalized derivation, derivation and centralizer on R, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study ad-nilpotent elements of semiprime rings R with involution ∗ whose indices of ad-nilpotence differ on Skew (R , ∗) and R. The existence of such an ad-nilpotent element a implies the existence of a GPI of R, and determines a big part of its structure. When moving to the symmetric Martindale ring of quotients Q m s (R) of R, a remains ad-nilpotent of the original indices in Skew (Q m s (R) , ∗) and Q m s (R) . There exists an idempotent e ∈ Q m s (R) that orthogonally decomposes a = e a + (1 - e) a and either ea and (1 - e) a are ad-nilpotent of the same index (in this case the index of ad-nilpotence of a in Skew (Q m s (R) , ∗) is congruent with 0 modulo 4), or ea and (1 - e) a have different indices of ad-nilpotence (in this case the index of ad-nilpotence of a in Skew (Q m s (R) , ∗) is congruent with 3 modulo 4). Furthermore, we show that Q m s (R) has a finite Z -grading induced by a ∗ -complete family of orthogonal idempotents and that e Q m s (R) e , which contains ea, is isomorphic to a ring of matrices over its extended centroid. All this information is used to produce examples of these types of ad-nilpotent elements for any possible index of ad-nilpotence n. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, F a generalized skew derivation of R, L a non-central Lie ideal of R and n ≥ 1 a fixed integer such that [ F (x) , y ] n = [ x , F (y) ] n , for all x , y ∈ L . Then there exists λ ∈ C such that F (x) = λ x , for any x ∈ R , unless R is an order in a 4-dimensional central simple algebra. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring with center Z (R) and with extended centroid C and let σ : R → R be an automorphism. Assume that τ : R → R is an anti-homomorphism, such that the image of τ has small centralizer. It is proved that the following are equivalent: (1) x σ x τ = x τ x σ for all x ∈ R ; (2) x σ + x τ ∈ Z (R) for all x ∈ R ; (3) x σ x τ ∈ Z (R) for all x ∈ R. In this case, there exists an idempotent e ∈ C , such that (1 - e) R is a commutative ring and the semiprime ring e R is equipped with an involution τ ~ , which is induced canonically by τ . Note that one can easily obtained the main result in Lee (Commun Algebra 46(3):1060–1065, 2018) when σ = i d R. [ABSTRACT FROM AUTHOR]

In this paper we study additive maps satisfying some identities and investigate the following: Let R be a semiprime ring, (α1, α2) be automor-phisms on R and Δ; δ be additive maps from R to R. If Δ and δ satisfying any one of the following identities (i) Δ(xnyn) = Δ(xn)α1(yn) + α2(xn)Δ(yn) (ii) Δ(xn) = Δ(xn-m)α1(xm) + α2(xn-m)δ(xm), for all x; y 2 R; then Δ will be a generalized ((α1, α2)-derivation with associated ((α1, α2)-derivation δ on R. [ABSTRACT FROM AUTHOR]

In this paper, we study ad-nilpotent elements in Lie algebras arising from semiprime associative rings R free of 2-torsion. With the idea of keeping under control the torsion of R, we introduce a more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a only technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows us to be more precise when setting the torsion inside the ring R in order to describe its ad-nilpotent elements. If R is a semiprime ring and a ∈ R is a pure ad-nilpotent element of R of index n with R free of t and n t -torsion for t = [ n + 1 2 ] , then n is odd and there exists λ ∈ C (R) such that a - λ is nilpotent of index t. If R is a semiprime ring with involution ∗ and a is a pure ad-nilpotent element of Skew (R , ∗) free of t and n t -torsion for t = [ n + 1 2 ] , then either a is an ad-nilpotent element of R of the same index n (this may occur if n ≡ 1 , 3 (mod 4) ) or R is a nilpotent element of R of index t + 1 , and R satisfies a nontrivial GPI (this may occur if n ≡ 0 , 3 (mod 4) ). The case n ≡ 2 (mod 4) is not possible. [ABSTRACT FROM AUTHOR]

The main purpose of thess notes investigated some certain properties and relation between higher derivation (HD,for short) and Lie ideal of semiprime rings and prime rings,we gave some results about that.

Let d be a derivation of a semiprime ring R and L a nonzero Lie ideal of R. In this note, it is proved that every noncentral square-closed Lie ideal of R contains a nonzero ideal of R. Further, we use this result to characterize the conditions: d(xy) = d(x)d(y), d(xy) = d(y)d(x) on L. With this, a theorem of Ali et al. [14] can be deduced.

The present paper investigates some properties of generalized reverse derivations on prime and semiprime rings. Firstly, the commutativity of a prime ring R is examined under the following differential identities provided by a generalized reverse derivation F associated with a reverse derivation d of R on a one-sided ideal of R and a mapping G: (i) F([x,y]) ∓ [G(z),y] ∈ Z(R), (ii) F([x,y]) ∓ G(z) o y ∈ Z(R), (iii) F([x,y]) ∓ G(z)x ∈ Z(R), (iv) F ([x, y]) ∈ xG(z) ∈ Z (R), (v) F ([x,y]) ∓ xz ∈ Z (R), (vi) F (yx) ∈ [G(z),y] ∈ Z (R), (vii) F (yx) ∓ G(z)oy ∈ Z (R), (viii) F (xy) ∓ G(z)x ∈ Z (R), (ix) F (xy) ∓ xG(z) ∈ Z (R), (x) F (y o x) ∓ [G(z),y] ∈ Z (R), (xi) F(yox) ∓ G(z)oy ∈ Z(R), (xii) F(xoy) ∓ G(z)x ∈ Z(R), (xiii) F (x o y) ∓ xG(z) ∈ Z (R), (xiv) F (x o y) ∓ xz ∈ Z (R). Secondly, we study the relationships between r--generalized reverse derivations and l--generalized derivation and l--generalized reverse derivation and r--generalized derivations on a noncentral square-closed Lie ideal in a semiprime ring. Finally, we provide two open problems. [ABSTRACT FROM AUTHOR]

Let (d i) i ∈ N and (δ j) j ∈ N be two higher derivations of order m and n respectively on semiprime ring R such that d m (x) x - x δ n (x) ∈ Z (R) for all x ∈ I , where I be an ideal of R. In this article we generalize Posner's second theorem for higher derivation and prove that either R is commutative or some linear combination of (δ j) sends Z(R) to zero. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to describe the structure of a pair of additive mappings that are commuting on a semiprime ring. Furthermore, we prove that the existence of different commuting epimorphisms on a prime ring forces the ring to be commutative. Finally, we characterize additive mappings, which act as homomorphisms or antihomomorphisms on a semiprime ring. [ABSTRACT FROM AUTHOR]

Let ℛ be a prime ring with char (ℛ) ≠ 2 , ℒ a non-central Lie ideal of ℛ , 𝒬 its Martindale quotient ring and 𝒞 its extended centroid. Let F : ℛ → ℛ and G : ℛ → ℛ be nonzero generalized derivations on ℛ such that [ F (x) , F (y) ] k = G ([ x , y ] k) , ∀ x , y ∈ ℒ. Then there exists λ ∈ 𝒞 such that F (x) = λ x and G (x) = λ k + 1 x , for any x ∈ ℛ , unless ℛ ⊆ M 2 (𝒞 ¯) , where 𝒞 ¯ is the algebraic closure of 𝒞. [ABSTRACT FROM AUTHOR]

In this paper, we give an affirmative answer to two conjectures on generalized (m, n)-Jordan derivations and generalized (m, n)-Jordan centralizers raised in Ali and Fošner (Algebra Colloq 21:411–420, 2014) and Fošner (Demonstr Math 46:254–262, 2013). Precisely, when R is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized (m, n)-Jordan derivation (resp., a generalized (m, n)-Jordan centralizer) is a derivation (resp., a two-sided centralizer). [ABSTRACT FROM AUTHOR]

The main aim of this manuscript is to prove the following result. Let n > 2 be a fixed integer and R be a k-torsion free semiprime ring with identity, where k ∈ {2, n - 1, n}. Let us assume that for the additive mapping T : R→ R 3T(xn) = T(x)xn-1+ xT(xn-2)x + xn-1T(x), x ∈R, is also fulfilled. Then T is a two-sided centralizer.

In this article, we study multiplicative (generalized)-skew derivation G and multiplicative left centralizer H satisfying certain conditions in semiprime rings. Moreover, some examples are given to demonstrate that the semiprimeness imposed on the hypotheses of various theorems is essential.

Let $R$ be a ring with center $Z(R)$. A mapping $F:R\rightarrow R$ is called a multiplicative generalized derivation, if $F(xy)=F(x)y+xg(y)$ is fulfilled for all $x,y\in R$, where $g:R\rightarrow R$ is a derivation. In the present paper, our main object is to study the situations: (1) $F(xy)- F(x)F(y)\in Z(R)$, (2) $F(xy)+ F(x)F(y)\in Z(R)$, (3) $F(xy)- F(y)F(x)\in Z(R)$, (4) $F(xy)+ F(y)F(x)\in Z(R)$, (5) $F(xy)- g(y)F(x)\in Z(R)$; for all $x,y$ in some suitable subset of $R$.

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Let R be a semiprime ring, not necessarily with unity, and a , b ∈ R. Let I (a) (respectively, Ref (a)) denote the set of inner (respectively, reflexive) inverses of a in R. It is proved that if I (a) ∩ I (b) ≠ ∅ , then I (a) ⊆ I (b) if and only if b = a w b = b w a for all w ∈ I (a). As an immediate consequence, if ∅ ≠ I (a) = I (b) , then a = b (see Theorem 7 in [A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl.18(7) (2019) 1950128] for rings with unity). We also give a generalization of Theorem 10 in [A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl.18(7) (2019) 1950128] by proving that if ∅ ≠ Ref (a) ⊆ Ref (b) then a = b. [ABSTRACT FROM AUTHOR]