Joso Vukman

On left derivations and related mappings

Let R be a ring and X be a left R-module. The purpose of this paper is to investigate additive mappings D1: R -* X and D2: R -X that satisfy D, (ab) = aD1 (b) + bD, (a), a, b E R (left derivation) and D2(a2) = 2aD2(a), a E R (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of R into X implies R is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.

Symmetric bi-derivations on prime and semi-prime rings

SummaryLetR be a ring. A bi-additive symmetric mappingD(.,.): R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, a mappingx ↦ D(x, y) is a derivation. The purpose of this paper is to prove some results concerning symmetric bi-derivations on prime and semi-prime rings. We prove that the existence of a nonzero symmetric bi-derivationD(.,.): R × R → R, whereR is a prime ring of characteristic not two, with the propertyD(x, x)x = xD(x, x), x ∈ R, forcesR to be commutative. A theorem in the spirit of a classical result first proved by E. Posner, which states that, ifR is a prime ring of characteristic not two andD1,D2 are nonzero derivations onR, then the mappingx ↦ D1(D2(x)) cannot be a derivation, is also presented.

On derivations in prime rings and Banach algebras

Let R be a ring with center Z(R). A mapping F: R —> R is said to be centralizing on R if (F(x), x) e Z(R) holds for all x £ R. The main purpose of this paper is to prove the following result, which generalizes a classical result of Posner: Let R be a prime ring of characteristic not 2, 3, and 5. Suppose there exists a nonzero derivation D: R —» R , such that the mapping x t*-» ((D(x) , x), x) is centralizing on R. In this case R is commu- tative. Combining this result with some well-known deep results of Sinclair and Johnson, we generalize Yoods noncommutative extension of the Singer-Wermer theorem. Preliminaries This paper is a continuation of our earlier work (17). Throughout, R repre- sents an associative ring with center Z(R). We write (x, y) for xy -yx , and use the identities (xy, z) = (x, z)y + x(y, z), (x, yz) = (x , y)z + y(x , z). Recall that R is prime if aRb = (0) implies that either a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping D from R to R is called a derivation if D{xy) - D{x)y + x D{y) holds for all x, y £ R. A derivation D is inner if there exists a £ R, such that D{x) — (a , x) holds for all x £ R. An additive mapping D from R to R is called a Jordan derivation if D{x2) - D{x)x + xD{x) holds for all x £ R. Obviously, every derivation is a Jordan derivation. The converse is in general not true. Herstein (7) has proved that every Jordan derivation on a prime ring of characteristic not two is a derivation. A brief proof of Hersteins result can be found in (4). Cursack (6) has generalized Hersteins result on 2-torsionfree (i.e., such that 2x = 0 implies x = 0) semiprime rings (see also (2)). A mapping F from R to R is said to be commuting on R if (F{x), x) = 0 holds for all x £ R, and is said to be centralizing on R if (F{x), x) £ Z{R) holds for all x £ R. For results concerning commuting, centralizing, and related mappings in prime and semiprime rings, we refer to (1, 5, 9, 10, 16, 17) where further references can be found.

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.

Two results concerning symmetric bi-derivations on prime rings

SummaryLetR be a ring. A bi-additive symmetric mappingD(.,.): R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, the mappingx → D(x, y) is a derivation. The purpose of this paper is to prove two results concerning symmetric bi-derivations on prime rings. The first result states that, ifD1 andD2 are symmetric bi-derivations on a prime ring of characteristic different from two and three such thatD1(x, x)D2(x,x) = 0 holds for allx ∈ R, then eitherD1 = 0 orD2 = 0. The second result proves that the existence of a nonzero symmetric bi-derivation on a prime ring of characteristic different from two and three, such that [[D(x, x),x],x] ∈ Z(R) holds for allx ∈ R, whereZ(R) denotes the center ofR, forcesR to be commutative.

Some results concerning additive mappings and derivations on semiprime rings

Let R be a 2-torsion free semiprime ring and let f : R ! R be an additive mapping satisfying the relation [f(x); x2] = 0 for all x 2 R. We prove that in this case [f(x); x] = 0 holds for all x 2 R. This result makes it possible to prove the following result. Let R be a 2-torsion free semiprime ring and let D;G : R ! R be derivations. Suppose that the relation [D2(x) + G(x); x2] = 0 holds for all x 2 R. Then D and G both map R into its center.

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.

Identities with derivations and automorphisms on semiprime rings

The purpose of this paper is to investigate identities with derivations and automorphisms on semiprime rings. A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Mayne proved that in case there exists a nontrivial centralizing automorphism on a prime ring, then the ring is commutative. In this paper, some results related to Posners theorem as well as to Maynes theorem are proved.

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

An equation related to two-sided centralizers in prime rings

In this paper we prove the following result: Let R be a prime ring and let T : R ? R be an additive mapping satisfying the relation nT(xn)=T(x)xn-1 + xT(x)xn-2 + ... + xn-1T(x) for all x in R where n > 1 is some fixed integer. If char(R) = 0 or n = char(R) ? 2, then T is of the form T(x) = ?x for all x in R and some fixed element ? in R where C is the extended centroid of R.