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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]

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]

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]

Let R be an associative ring. A *- semimultiplier is an additive map F : R = R such that F(xy) = F(x)g(y8) = g(x*)F(y) where g is some additive map and F(g(x)) = g(F(x)) for all x 2 R . We make extensive use of functional identities defined in prime ring R of the forms xE1(y) + yE2(x) = 0 or xE1(y) + yE2(x) 2 Z(R) = C where E1, E2 are any arbitrary functions on the prime ring R and Z(R), C are the center and the extended centroid of R respectively. We have proved that in a prime ring R under some additional conditions, a 8 - semimultiplier F : R = R is a map given by F(x) = x + (x), where * 2 C and : R = C. We have also shown that a prime ring admitting the *-semimultiplier satisfies S4, the standard identity of degree 4 under some suitable conditions. Further, some other important results are also incorporated. [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]

We present an historical account of the study of derivations, generalized deriva-tions, n-derivations, generalized n-derivation and other kinds of derivations in near-rings, based on the work of several authors. Moreover, recent results on semigroup ideals and generalized n-derivations on these topics have been discussed in details. Examples of various notions have also been included. [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]

In this paper, we introduce the notion of generalized n-derivation in near-ring N and investigate several identities involving generalized n-derivations of a prime near-ring N which force N to be a commutative ring. Finally some more related results are also obtained. [ABSTRACT FROM AUTHOR]