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Remarks on Generalized Derivations in Prime and Semiprime Rings
Get PDFLet R be a ring with center Z and I a nonzero ideal of R. An additive mapping F:R→R is called a generalized derivation of R if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) for all x,y∈R. In the present paper, we prove that if F([x,y])=±[x,y] for all x,y∈I or F(x∘y)=±(x∘y) for all x,y∈I, then the semiprime ring R must contains a nonzero central ideal, provided d(I)≠0. In case R is prime ring, R must be commutative, provided d≠0. The cases (i) F([x,y])±[x,y]∈Z and (ii) F(x∘y)±(x∘y)∈Z for all x,y∈I are also studied
Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings
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Some identities involving multiplicative generalized derivations in orime and semiprime rings
No full textLet be a ring with center . A mapping is called a multiplicative generalized derivation, if is fulfilled for all , where is a derivation. In the present paper, our main object is to study the situations: (1) , (2) , (3) , (4) , (5) ; for all in some suitable subset of .</jats:p
Derivations with Engel conditions on multilinear polynomials in prime rings
No full textAbstractLet</jats:p
Vanishing Power Values of Commutators with Derivations on Prime Rings
Get PDFLet R be a prime ring of char R≠2, d a nonzero derivation of R and ρ a nonzero right ideal of R such that [[d(x),x]n,[y,d(y)]m]t=0 for all x,y∈ρ, where n≥0, m≥0, t≥1 are fixed integers. If [ρ,ρ]ρ≠0, then d(ρ)ρ=0
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