RENDICONTI DEL CIRCOLO MATE~ATICO DI PALERMO Sede II, Tomo XXXIII (1984), pp. 441-455 ON THE COMPOSITION OF DERIVATIONS 1. KREMPA -- 1. MATCZUK Posner ([9]) has shown that for any prime ring R of characteristic different from 2 the composition of any two non-zero derivations is not a derivation. On the other hand, it is well known ([4]) that if char R=n for a prime number n and d is a derivation of R, then dn is also a derivation. Our main objective is to extend the above mentioned result of Posner in the ease of commutative domains, and to apply this results to the investigation of con- nections either between derivations and a center, or between derivations and a generalized centroid of a prime ring. For this purpose, we are first going to introduce a method of notation for the composition of derivations which, we hope, will also be useful in other situations. Introduction Let R denote an associative ring. For convenience, we will take char R = oo if the additive group of R is torsion-free. For a derivation d of R we will take d~ even if d=0. Let dt ..... d, be derivations of R and U={1 ..... n}. For any subset V of U we will put I 1 if i6V (1) dv (x) = dl',.. . d~, (x) where Ei = 0 ifi6V. In particular, we get dv(x)=dl...d,(x), d~(x)=x and if V={i}, then dv (x)= dl (x). Further, by writing V-VIU...UVr we will understand that V is a sum of disjoint subsets Vi.