Basudeb Dhara

Remarks on Generalized Derivations in Prime and Semiprime Rings

Let be a ring with center and a nonzero ideal of . An additive mapping is called a generalized derivation of if there exists a derivation such that for all . In the present paper, we prove that if for all or for all , then the semiprime ring must contains a nonzero central ideal, provided . In case is prime ring, must be commutative, provided . The cases (i) and (ii) for all are also studied.

Power Values of Derivations with Annihilator Conditions on Lie Ideals in Prime Rings

Let R be a prime ring of char R ≠ 2, d a nonzero derivation of R, U a noncentral Lie ideal of R, and a ∈ R. If au n 1 d(u) n 2 u n 3 d(u) n 4 u n 5 … d(u) n k−1 u n k = 0 for all u ∈ U, where n 1, n 2,…,n k are fixed non-negative integers not all zero, then a = 0 and if a(u s d(u)u t ) n ∈ Z(R) for all u ∈ U, where s ≥ 0, t ≥ 0, n ≥ 1 are some fixed integers, then either a = 0 or R satisfies S 4, the standard identity in four variables.

NOTES ON GENERALIZED DERIVATIONS ON LIE IDEALS IN PRIME RINGS

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that u s H(u)u t = 0 for all u 2 L, where s ‚ 0,t ‚ 0 are fixed integers. Then H(x) = 0 for all x 2 R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x,y 2 R, the commutator xyiyx will be denoted by (x,y). An additive mapping d from R to R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x,y 2 R. A derivation d is inner if there exists a 2 R such that d(x) = (a,x) holds for all x 2 R. An additive subgroup L of R is said to be a Lie ideal of R if (u,r) 2 L for all u 2 L, r 2 R. The Lie ideal L is said to be noncommutative if (L,L) 6 0. Hvala (8) introduced the notion of generalized derivation in rings. An additive mapping H from R to R is called a generalized derivation if there exists a derivation d from R to R such that H(xy) = H(x)y+xd(y) holds for all x,y 2 R. Thus the generalized derivation covers both the concepts of derivation and left multiplier mapping. The left multiplier mapping means an additive mapping F from R to R satisfying F(xy) = F(x)y for all x,y 2 R. Throughout this paper R will always present a prime ring with center Z(R), extended centroid C and U its Utumi quotient ring. It is well known that if ‰ is a right ideal of R such that u n = 0 for all u 2 ‰, where n is a fixed positive integer, then ‰ = 0 (7, Lemma 1.1). In (2), Chang and Lin consider the situation when d(u)u n = 0 for all u 2 ‰ and u n d(u) = 0 for all u 2 ‰, where ‰ is a nonzero right ideal of R. More precisely, they proved the following: Let R be a prime ring, ‰ a nonzero right ideal of R, d a derivation of R and n a fixed positive integer. If d(u)u n = 0 for all u 2 ‰, then d(‰)‰ = 0 and if u n d(u) = 0 for all u 2 ‰, then d = 0 unless R » M2(F), the 2◊2 matrices over a field F of two elements.

ON n-CENTRALIZING GENERALIZED DERIVATIONS IN SEMIPRIME RINGS WITH APPLICATIONS TO C*-ALGEBRAS

Let R be a ring with center Z(R) and n be a fixed positive integer. A mapping f : R → R is said to be n-centralizing on a subset S of R if f(x)xn – xn f(x) ∈ Z(R) holds for all x ∈ S. The main result of this paper states that every n-centralizing generalized derivation F on a (n + 1)!-torsion free semiprime ring is n-commuting. Further, we prove that if a generalized derivation F : R → R is n-centralizing on a nonzero left ideal λ, then either R contains a nonzero central ideal or λD(Z) ⊆ Z(R) for some derivation D of R. As an application, n-centralizing generalized derivations of C*-algebras are characterized.

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x1,…, xn) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) − f(r)G(f(r))) = 0 for all r = (r1,…, rn) ∈ Rn, then one of the following holds: There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a − q] = 0 and f(x1,…, xn)2 is central valued on R; There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R; There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa − bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C; R satisfies s4 and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa − bx and G(x) = ax + xb for all x ∈ R; There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb − δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x1,…, xn)2 is central valued on R.

Co-commutators with generalized derivations in prime and semiprime rings

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F and G two nonzero generalized derivations of R, I an ideal of R and f(x1, . . . , xn) be a multilinear polynomial over C which is not central valued on R. If F (f(x1, . . . , xn))f(x1, . . . , xn)− f(x1, . . . , xn)G(f(x1, . . . , xn)) = 0 for all x1, . . . , xn ∈ I, then one of the following holds: (1) F (x) = xa and G(x) = xb for all x ∈ R with a = b ∈ C; (2) F (x) = xa and G(x) = bx for all x ∈ R with a = b; (3) F (x) = ax and G(x) = xb for all x ∈ R with a = b ∈ C; (4) F (x) = ax and G(x) = xb for all x ∈ R with a = b and f(x1, . . . , xn) is central valued on R; (5) F (x) = ax and G(x) = bx for all x ∈ R, with a = b ∈ C. We finally extend the result to a semiprime ring R in case F (x)x − xG(x) = 0 for all x ∈ R. Throughout this paperR always denotes an associative prime ring with center Z(R), extended centroid C and U its Utumi quotient ring. The Lie commutator of x and y is denoted by [x, y] and defined by [x, y] = xy − yx for x, y ∈ R. An Mathematics Subject Classification: 16N60, 16W25 .

Generalized Derivations and Left Ideals in Prime and Semiprime Rings

Let 𝑅 be an associative ring, 𝜆 a nonzero left ideal of 𝑅, 𝑑∶𝑅→𝑅 a derivation and 𝐺∶𝑅→𝑅 a generalized derivation. In this paper, we study the following situations in prime and semiprime rings: (1) 𝐺(𝑥∘𝑦)=𝑎(xy±yx); (2) 𝐺[𝑥,𝑦]=𝑎(xy±yx); (3) 𝑑(𝑥)∘𝑑(𝑦)=𝑎(xy±yx); for all 𝑥,𝑦∈𝜆 and 𝑎∈{0,1,−1}.

Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings

Let R be an associative prime ring of char R 6= 2 with center Z(R) and extended centroid C, f(x1, . . . , xn) a nonzero multilinear polynomial over C in n noncommuting variables, d a nonzero derivation of R and ρ a nonzero right ideal of R. We prove that: (i) if [d(f(x1, . . . , xn)), f(x1, . . . , xn)] = 0 for all x1, . . . , xn ∈ ρ then ρC = eRC for some idempotent element e in the socle of RC and f(x1, . . . , xn) is central-valued in eRCe unless d is an inner derivation induced by b ∈ Q such that b = 0 and bρ = 0; (ii) if [d(f(x1, . . . , xn)), f(x1, . . . , xn)] ∈ Z(R) for all x1, . . . , xn ∈ ρ then ρC = eRC for some idempotent element e in the socle of RC and either f(x1, . . . , xn) is central in eRCe or eRCe satisfies the standard identity S4(x1, x2, x3, x4) unless d is an inner derivation induced by b ∈ Q such that b = 0 and bρ = 0. Mathematics Subject Classification: 16W25, 16R50, 16N60.

Generalized derivations acting on multilinear polynomials in prime rings

AbstractLet R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x1,..., xn) a multilinear polynomial over C which is not central valued on R. If