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Content available Semiprime FDI-rings
In this paper we present some results for FDI-rings, i.e. rings with a complete set of pairwise orthogonal primitive idempotents. We consider the nilpotency index of ideals and give its upper band for ideals in some classes of rings. We also give a new proof of a criterion of semiprime FDI-rings to be prime.
Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R, f(x1,…,xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(f(r))f(r) – f(r)H(f(r)) € C, for all r = (r1,…,rn) € In. Then one of the following holds: 1. there exist a; b; p (…) ; 2. R satisfies s4, the standard identity of degree 4, and there exist a; (…) ; 3. R satisfies s4 and there exist a; (…) ; 4. R satisfies s4 and there exist a; (…) ; 5. there exists e2 = e (…) and one of the following holds: (a) (…) is an identity for I; (b) char(R) = 2 and s4(x1, x2, x3, x4)x5 is an identity for I; (c) (…) is an identity for I and there exist a, a’, b, b’ (…) , a derivation of R, such that G(x) = ax + xa’ + d(x), H(x) = bx + xb’ – d(x), for all x (…) R, with (a – b’ – α)I = (0) = (b – a’ – α)I.
Content available remote Derivation with engel conditions on multilinear polynomials in prime rings
Let R be a prime ring with extended centroid C and characteristic different from 2, d a nonzero derivation of R, f(x1,..., xn) a nonzero multilinear polynomial over C such that [d2(f(x1,... , xn)),d(f(x1,... ,xn))]k = 0 for all x1,... ,xn in some nonzero right ideal [...] of R, where k is a fixed positive integer. If d(p) p [..]0, then pC = eRC for some idempotent e in the socle of RC and /(x1,..., xn) is central-valued on eRCe.
Content available remote On left multipliers and the commutativity of prime rings
Let R be an associative ring. An additive mapping H : R —> R is called a left multiplier if H(xy) = H(x)y, holds for all x, y e R. In this paper, we investigate commutativity of prime rings satisfying certain identities involving left multiplier. Some related results have also been discussed.
Content available remote On semiderivations of prime rings
Content available remote On Jordan ideals and Jordan derivations of prime rings
Let R be a 2-torsion free prime ring, and let J be a nonzero Jordan ideal and a subring of R. In the present paper it is shown that if d is an additive mapping of R into itself satisfying d(u2) = d(u)u + ud(u), for all u 6 J, then d(uv) = d(u)v + ud(v), for all u, v J.
Content available remote Centralizing mappings and derivations on semiprime rings
In this paper we study some properties of centralizing mappings on semi-prime rings. The main purpose is to prove the result: Let -R be a semiprime ring and f an endomorphism of R, g an epimorphism of R such that the mapping x -> [f(x),g(x)] is central. Then [f(x),g(x)] = 0 holds for all x e R. We also establish some results about (alpha,beta)-derivations.
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