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1

Semiprime FDI-rings

Gubareni N.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 4

49--59

EN

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.

2

Cocentralizing generalized derivations on multilinear polynomial on right ideals of prime rings

Filippis V. de, Dhara B.

Demonstratio Mathematica

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2014

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Vol. 47, nr 1

22--36

EN

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.

3

Derivation with engel conditions on multilinear polynomials in prime rings

Dhara B.

Demonstratio Mathematica

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2009

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Vol. 42, nr 3

467-478

EN

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.

4

On (m, n)-Jordan derivations and commutativity of prime rings

Vukman J.

Demonstratio Mathematica

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2008

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Vol. 41, nr 4

773-778

5

On left multipliers and the commutativity of prime rings

Ashraf M., Ali S.

Demonstratio Mathematica

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2008

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Vol. 41, nr 4

763-771

EN

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.

6

An identity with derivations on rings and Banach algebras

Fosner A., Fosner M., Vukman J.

Demonstratio Mathematica

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2008

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Vol. 41, nr 3

525-530

7

On semiderivations of prime rings

Ashrat M., Rehman N.

Demonstratio Mathematica

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2004

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Vol. 37, nr 2

275--284

8

On Jordan ideals and Jordan derivations of prime rings

Ashrat M., Rehman N.

Demonstratio Mathematica

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2004

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Vol. 37, nr 3

517--523

EN

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.

9

Centralizing mappings and derivations on semiprime rings

Chaudhry M., Thaheem A.

Demonstratio Mathematica

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2004

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Vol. 37, nr 2

285--292

EN

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.