Irena Kosi-Ulbl

On some equations related to derivations in rings

Let m and n be positive integers with m + n ≠ 0 , and let R be an ( m + n + 2 ) ! -torsion free semiprime ring with identity element. Suppose there exists an additive mapping D : R → R , such that D ( x m + n + 1 ) = ( m + n + 1 ) x m D ( x ) x n is fulfilled for all x ∈ R , then D is a derivation which maps R into its center.

On certain equations in rings

In this paper we prove the following result: Let R be a 2-torsion free semiprime ring. Suppose there exists an additive mapping T : R → R such that T(xyx) = T(x)yx − xT(y)x + xyT(x) holds for all pairs x, y ∈ R . Then T is of the form 2T(x) = qx + xq , where q is a fixed element in the symmetric Martindale ring of quotients of R .

A note on derivations in semiprime rings

We prove in this note the following result. Let n>1 be an integer and let R be an n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D:R→R such that D(xn)=∑j=1nxn−jD(x)xj−1 is fulfilled for all x∈R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associative ring with center Z(R). Given an integer n>1, a ring R is said to be n-torsion-free if for x∈R, nx=0 implies that x=0. Recall that a ring R is prime if for a,b∈R, aRb=(0) implies that either a=0 or b=0, and is semiprime in case aRa=(0) implies that a=0. An additive mapping D:R→R is called a derivation if D(xy)=D(x)y

Centralisers on rings and algebras

In this paper we investigate identities related to centralisers in rings and algebras. We prove, for example, the following result. Let A be a semisimple H * -algebra and let T : A → A be an additive mapping satisfying the relation T ( x m+n+1 ) = x m T ( x ) x n for all x ∈ A and some integers m ≥ 1, n ≥ 1. In this case T is a left and a right centraliser.

On dependent elements in rings

Let R be an associative ring. An element a∈R is said to be dependent on a mapping F:R→R in case F(x)a=ax holds for all x∈R. In this paper, elements dependent on certain mappings on prime and semiprime rings are investigated. We prove, for example, that in case we have a semiprime ring R, there are no nonzero elements which are dependent on the mapping α

On

In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be fixed integers and let R be an mn(m+n)-torsion free semiprime ring. Suppose there exists an additive mapping T : R→ R satisfying the relation (m + n)T (x) = mT (x)x + nxT (x) for all x ∈ R ((m,n)-Jordan centralizer). In this case T is a two-sided centralizer. Throughout, R will represent an associative ring with center Z(R). As usual, the commutator xy − yx will be denoted by [x, y]. We shall use the commutator identity [xy, z] = [x, z]y + x[y, z]. Given an integer n ≥ 2, a ring R is said to be n-torsion free, if for x ∈ R, nx = 0 implies x = 0. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. We denote by char(R) the characteristic of a prime ring R. We denote by Qr and C Martindale right ring of quotients and extended centroid of a semiprime ring R, respectively. For explanation of Qr and C, we refer the reader to [2]. An additive mapping T : R→ R is called a left centralizer in case T (xy) = T (x)y holds for all pairs x, y ∈ R and is called a left Jordan centralizer in case T (x) = T (x)x holds for all x ∈ R. In case R has the identity element, T : R → R is a left centralizer iff T is of the form T (x) = ax for all x ∈ R, where a ∈ R is a fixed element. For a semiprime ring R all left centralizers are of the form T (x) = qx for all x ∈ R, where q is a fixed element from Qr (see [2, Chapter 2]). The definition of right centralizer and right Jordan centralizer should be self-explanatory. We call T : R → R a two-sided centralizer in case T is both a left and a right centralizer. In case T : R→ R is a two-sided centralizer, where R is a semiprime ring with extended centroid C, then there exists element Mathematics Subject Classification: 16W10, 39B05.

(m,n)

En este articulo nosotros provamos el seguiente resultado. Sea X un espacio de Banach real o complejo, sea L(X) a algebra de todos los operadores linares acotados sobre X, y sea una algebra de operadores estandar. Suponga una aplicacion lineal verificando la relacion . En este caso D es de la forma y algun , lo que significa que D es una deriviacion lineal. En particual, D es continua. Nosotros aplicamos este resultado el cual generaliza un resultado clasico de Chernoff, para H*-algebras semisimple. Este trabajo fue motivado por un trabajo de Herstein [4], Chernoff [2] y Molnar [5] y este una continuacion de nuestro reciente trabajo [8] y [9].

-Jordan centralizers of semiprime rings

In this paper we investigate identities with two automorphisms on semiprime rings. We prove the following result: Let T, S : R —> R be automorphisms where R is a 2-torsion free semiprime ring satisfying the relation T(x)x = xS(x) for all x € R. In this case the mapping x >—> T(x ) — x maps R into its center and T = S. 1. Preliminaries Throughout, R will represent an associative ring with center Z(R). A ring R is n-torsion free, where n > 1 is an integer, in case nx = 0, x G R implies x = 0. As usual the commutator xy — yx will be denoted by [x,y\. We shall frequently use the commutator identities [xy, z] = [x, z] y + x [y, z] and [x, yz] = [x,y]z + y [x, z]. We denote by I the identity mapping on a ring R. Recall that R is prime if aRb = (0) implies a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping D : R —> R, where R is an arbitrary ring, is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x,y £ R. We denote by C the extended centroid of a semiprime ring R and by Q Martindale ring of quotients. For the explanation of the extended centroid of a semiprime ring R and the Martindale ring of quotients we refer the reader to [1]. A mapping / : R —> R is called centralizing on R if [f(x), x] e Z(R) holds for all x G R; in the special case when [f(x), x] = 0 holds for all x e R, the mapping / is said to be commuting on R. The history of commuting and centralizing mappings goes back to 1955 when Divinsky [6] proved that a simple Artinian ring is commutative if it has a commuting nontrivial automorphism. Two years later Posner [9] has proved that the existence of a nonzero centralizing derivation on a prime ring forces the ring 2000 Mathematics Subject Classification: 16E99, 08A35.