Let R be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when R is commutative, is provided. If, however, R has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if R is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If R is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of a is greater than $2$.

Let θ be a Jordan homomorphism from an algebra A into an algebra B. We find various conditions under which the restriction of θ to the commutator ideal of A is the sum of a homomorphism and an antihomomorphism. Algebraic results, obtained in the first part of the paper, are applied to the second part dealing with the case where A and B are C*-algebras.

In certain rings containing non-central idempotents we characterize homomorphisms, derivations, and multipliers by their actions on elements satisfying some special conditions. For example, we consider the condition that an additive map $h$ between rings $\mathcal{A}$ and $\mathcal{B}$ satisfies $h(x)h(y)h(z)=0$ whenever $x,y,z\in\mathcal{A}$ are such that $xy=yz=0$. As an application, we obtain some new results on local derivations and local multipliers. In particular, we prove that if $\mathcal{A}$ is a prime ring containing a non-trivial idempotent, then every local derivation from $\mathcal{A}$ into itself is a derivation.

Let d be a Jordan derivation from a ring $\cal{A}$ into an $\cal{A}$-bimodule $\cal{M}$. Our main result shows that the restriction of d to the ideal of $\cal{A}$ generated by certain higher commutators of $\cal{A}$ is a derivation. This general statement is used for proving that under various additional conditions d must be a derivation on $\cal{A}$. Furthermore, several examples of proper Jordan derivations are given, $C^{\ast}$-algebras admitting proper additive jordan derivations are characterized, and the connections with the related problems on jordan homomorphisms and jordan $\cal{A}$-module homomorphisms are discussed.

We consider elementary operators on centrally closed prime algebras that are Lie (or Jordan) homomorphisms or commutativity preservers.

AMS 2000 Mathematics subject classification: Primary 16N60. Secondary 16R50; 47B47

For each $n\ge 4$ we construct a class of examples of a minimal $C$ -dependent set of $n$ automorphisms of a prime ring $R$ , where $C$ is the extended centroid of $R$ . For $n=4$ and $n=5$ it is shown that the preceding examples are completely general, whereas for $n=6$ an example is given which fails to enjoy any of the nice properties of the above example.

Let R be a prime ring of characteristic not 2. Automorphisms α and β of R satisfying α ≠ β, α ≠ β−1, and α + α−1 = β + β-1 are characterized. This result is an algebraic analogue of some results for operator algebras.

Over the last few years a number of results giving conditions on a derivation of a Banach algebra implying that its range is contained in the radical have been obtained (see survey articles of Mathieu[7] and Murphy [8]). If an algebra is semi-simple, these conditions, of course, imply that a derivation is zero. In this paper we consider inner derivations that are non-zero in general, but their ranges are rather special and ‘small’ in some sense.

Let R be a prime ring of characteristic not 2, C be the extended centroid of R, and f: R → R be an additive map. Suppose that [f(x), x2] = 0 for all x ∈ R. Then there exist λ ∈ C and an additive map ζ: R → C such that f(x) = λx + ζ(x) for all x ∈ R. In particular, if f(x)2 = x2 for all x ∈ R, then ζ = 0 and either λ = 1 or λ= -1.

In this paper we characterize maps f: R —> R where R is semiprime, f is additive, and [f(x),f(y)] = [x,y] for all x,y ∊ R. It is shown that f(x) = λx + ξ(x) where λ ∊ C, λ2 = 1, and ξ: R —> C is additive where C is the extended centroid of R.

Let H be a Hilbert space, dim H ≥ 3, and B(H) the algebra of all bounded linear operators on H. We characterize bijective linear mappings on B(H) that preserve normal operators.

A map θ: M —> N where M and N are rings is said to preserve commutativity in both directions if the elements a,b ∊ M commute if and only if θ(a) and θ(b) commute. In this paper we show that if M and N are von Neumann algebras with no central summands of type I 1 or I 2 and θ is a bijective additive map which preserves commutativity in both directions then θ(x) = cφ(x) +f(x) where c is an invertible element in ZN , the center of N, φ M —> N is a Jordan isomorphism of M onto N, and f is an additive map of M into ZN .

It is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.

A mapping f of a ring R into itself is called skew-commuting on a subset S of R if f(s)s + sf(s) = 0 for all s ∈ S. We prove two theorems which show that under rather mild assumptions a nonzero additive mapping cannot have this property. The first theorem asserts that if R is a prime ring of characteristic not 2, and f: R → R is an additive mapping which is skew-commuting on an ideal I of R, then f(I) = 0. The second theorem states that zero is the only additive mapping which is skew-commuting on a 2-torsion free semiprime ring.

In this paper we prove algebraic generalizations of some results of C. J. K. Batty and A. B. Thaheem, concerned with the identity α + α−1 = β + β−1 where α and β are automorphisms of a C*-algebra. The main result asserts that if automorphisms α, β of a semiprime ring R satisfy α + α-1 = β + β−1 then there exist invariant ideals U1, U2 and U3 of R such that Ui ∩ Uj = 0, i ≠ j, U1 ⊕ U2 ⊕ U3 is an essential ideal, α = β on U1, α = β−1 on U2, and α2 = β2 = α−2 on U3. Furthermore, if the annihilator of any ideal in R is a direct summand (in particular, if R is a von Neumann algebra), then U1 ⊕ U2 ⊕ U3 = R.

We describe Jordan homomorphisms and Jordan triple homomorphisms onto 2-torsion free semiprime rings in which the annihilator of any ideal is a direct summand.

A well-known theorem of E. Posner [10] states that if the composition d1d2 of derivations d1d2 of a prime ring A of characteristic not 2 is a derivation, then either d1 = 0 or d2 = 0. A number of authors have generalized this theorem in several ways (see e.g. [1], [2], and [5], where further references can be found). Under stronger assumptions when A is the algebra of all bounded linear operators on a Banach space (resp. Hilbert space), Posner's theorem was reproved in [3] (resp. [12]). Recently, M. Mathieu [8] extended Posner's theorem to arbitrary C*-algebras.