Commutators with finite spectrum
The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum σ([dx,d′x]) is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d,d′] is the inner derivation implemented by a.
Derivations in prime rings
We prove two theorems that are easily conjectured, namely: (1) In a prime ring of characteristics not 2, if the iterate of two derivations is a derivation, then one of them is zero; (2) If d is a derivation of a prime ring such that, for all elements a of the ring, ad(a) -d(a)a is central, then either the ring is commutative or d is zero. DEFINITION. A ring R is called prime if and only if xay= 0 for all aER implies x=O or y=O. From this definition it follows that no nonzero element of the centroid has nonzero kernel, so that we can divide by the prime p, unless px = 0 for all x in R, in which case we call R of characteristic p. A known result that will be often used throughout this paper is given in
Normed units in abelian group rings
We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).