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Let S be a unital prime ∗-ring containing a nontrivial symmetric idempotent and let m s (S) be the maximal symmetric ring of quotients of S. Using the technique of Peirce decomposition and the theory of functional identities, we prove that a map : S → m s (S) satisfies (u v ∗ + v u ∗) = (u) v ∗ + (v) u ∗ + u (v) ∗ + v (u) ∗ for all u , v ∈ S if and only if is an additive ∗-derivation unless dim S = 4 and char(S) = 2. As an application, we shall characterize such maps in different operator algebras. [ABSTRACT FROM AUTHOR]

Let 풜 be a prime ring equipped with an involution ' * ' of order 2 and let m ≠ n be some fixed positive integers such that 풜 is 2 ⁢ m ⁢ n ⁢ (m + n) ⁢ | m - n | -torsion free. Let 풬 m ⁢ s ⁢ (풜) be the maximal symmetric ring of quotients of 풜 and consider the mappings ℱ and 풢 : 풜 → 풬 m ⁢ s ⁢ (풜) satisfying the relations (m + n) ⁢ ℱ ⁢ (a 2) = 2 ⁢ m ⁢ ℱ ⁢ (a) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ (a) and (m + n) ⁢ 풢 ⁢ (a 2) = 2 ⁢ m ⁢ 풢 ⁢ (a) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ (a) for all a ∈ 풜 . Using the theory of functional identities and the structure of involutions on matrix algebras, we prove that if ℱ and 풢 are additive, then 풢 = 0 . We also show that, in case ' * ' is any nonidentity anti-automorphism, the same conclusion holds if either ' * ' is not identity on 풵 ⁢ (풜) or 풜 is a PI-ring. [ABSTRACT FROM AUTHOR]

This work encompasses the study of certain weak maps termed as weak right η-centralizers defined on $ \mathfrak {D} $ D a noncommutative prime ring to $ \mathfrak {Q}_{ml}(\mathfrak {D}) $ Q ml (D). The first section provides an overview of the motivation behind our work. In the second section, our main focus is to establish the characterization of weak right η-centralizers while addressing the theory of functional identities, particularly in the challenging case of low dimensions. Furthermore, in the third section, we derive the characterization of weak Jordan σ-derivations based on the aforementioned characterization. [ABSTRACT FROM AUTHOR]

Let * be an involution of a non-commutative prime ring ℜ with the maximal symmetric ring of quotients and the extended centroid of ℜ denoted by Q m s (ℜ) and C , respectively. Consider : ℜ → Q m s (ℜ) be an additive map, if (u 2) − (u) u * − u (u) ∈ C for all u ∈ ℜ , then such a map is termed as a weak Jordan *-derivation. With the smart handling of the FI-theory and facing the challenging case of low dimensions, we prove that every weak Jordan *-derivation of ℜ is X -inner unless dim C ℜ C ≤ 9. Moreover, if * is of the first kind, then every weak Jordan *-derivation of ℜ is X -inner if and only if is Z (ℜ) -linear. [ABSTRACT FROM AUTHOR]

Consider ℜ as a prime ring which is non-commutative in structure with a suitable characteristic. Here, 풵 ⁢ (ℜ) is the center of ℜ and 풬 is the Utumi ring of quotients where 풞 is the extended centroid of ℜ . Suppose 풫 to be a Lie ideal of ℜ which is non-central. Let 풦 be a generalized derivation of ℜ related with derivation μ of ℜ . If 풦 satisfies certain typical algebraic identities, then we prove that 풦 is either the identity map or the zero map or the scalar map and further information is also drawn on the associated scalar unless ℜ embeds in M 2 ⁢ (풞) , a matrix ring of order 2 × 2 over 풞 . [ABSTRACT FROM AUTHOR]

Consider a ring with an (anti)-automorphism ∇ of finite order. The fundamental aim of this manuscript is to introduce the notions of ∇-(semi)prime ideal and ∇-(semi)prime ring as a generalization of the notions of (semi)prime ideal, ∗ -(semi)prime ideal, (semi)prime ring and ∗ -(semi)prime ring. Furthermore, we will investigate their basic properties and study the commutative property of ∇-prime rings which satisfy certain central differential identities on ∇-ideals. [ABSTRACT FROM AUTHOR]

Let R be an associative ring. A multiplicative semi-derivation d is a map on R satisfying d ⁢ (x ⁢ y) = d ⁢ (x) ⁢ g ⁢ (y) + x ⁢ d ⁢ (y) = d ⁢ (x) ⁢ y + g ⁢ (x) ⁢ d ⁢ (y) and d ⁢ (g ⁢ (x)) = g ⁢ (d ⁢ (x)) {d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)\quad\text{and}\quad d(g(x))=g(d(x))} for all x , y ∈ R {x,y\in R} , where g is any map on R. In this paper, we have obtained some conditions on R, which make d additive. Finally, we have also shown that every multiplicative semi-derivation on M n ⁢ (ℂ) {M_{n}(\mathbb{C})} , the algebra of all n × n {n\times n} matrices over the field ℂ {\mathbb{C}} of complex numbers, is an additive derivation. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of generalized ( σ , τ ) {(\sigma,\tau)} -n-derivations in a near-ring N and investigate a property involving generalized ( σ , τ ) {(\sigma,\tau)} -n-derivations of a prime near-ring N, which makes N a commutative ring. Additive commutativity of a prime near-ring N satisfying certain identities involving generalized ( σ , τ ) {(\sigma,\tau)} -n-derivations is also obtained. [ABSTRACT FROM AUTHOR]

Posner's first theorem states that if R is a prime ring of characteristic different from two, d1 and d2 are derivations on R such that the iterate d1d2 is also a derivation of R, then at least one of d1, d2 is zero. In the present paper we extend this result to *-prime rings of characteristic different from two. [ABSTRACT FROM AUTHOR]

In the present paper we investigate commutativity of *-prime ring R, which satisfies certain differential identities on *-ideals of R. Some results already known for prime rings on ideals have also been deduced. Finally, we provide several examples to justify that various restrictions imposed in the hypotheses of our theorems are not superfluous. [ABSTRACT FROM AUTHOR]

Let R be a *-ring. In this paper we introduce the notion of *-n-derivation in R. An additive mapping x → x* of R into itself is called an involution on R if it satisfies the conditions: (i) (x*)* = x, (ii) (xy)* = y*x* for all x, y ∊ R. A ring R equipped with an involution '*' is called a *-ring. It is shown that if a prime *-ring R admits a nonzero *-n-derivation (resp. a reverse *-n-derivation) D, then R is commutative. Further, some related properties of *-n-derivation in a semiprime *-ring have also been investigated. Finally, a structure theorem for *-n-derivation has been obtained. [ABSTRACT FROM AUTHOR]

A non-empty subset U of a near-ring N is said to be a semigroup left (resp. right) ideal of N if NU ⊆ U (resp. UN ⊆ U ) and if U is both a semigroup left ideal and a semigroup right ideal, it will be called a semigroup ideal. In the present paper, we investigate the commutativity of addition and multiplication of near-rings satisfying certain identities involving n -derivations on semigroup ideals and ideals. Furthermore, we study the conditions with semigroup ideals for n -derivations D 1 and D 2 of N which imply that D 1 = D 2 . [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of generalized multiplicative derivation in a near-ring N and investigate commutativity of 3-prime near-rings, showing that certain conditions involving generalized multiplicative derivations force N to be a commutative ring. Finally some more results related with the structure of these derivations are also obtained. [ABSTRACT FROM AUTHOR]