Remark on nilpotency of derivations (Q762594)

A derivation d on a ring R is called nilpotent if \(d^ nR=0\) for some positive integer n; the smallest such n is called the nilpotency of d. For semiprime R, the first and third authors had previously shown that the nilpotency of any nilpotent derivation must be odd if R is 2-torsion- free and a power of 2 if R has characteristic 2 [Can. Math. Bull. 26, 341-346 (1983; Zbl 0476.16028); Proc. Am. Math. Soc. 91, 357-358 (1984; Zbl 0519.16023)]. The present note improves the first result in the case of prime characteristic; specifically, if R is semiprime and of prime characteristic p, then the nilpotency of any nilpotent derivation must be of form \(n=a_ Lp^ L+a_{L+1}p^{L+1}+...+a_ Mp^ M\), where \(0\leq L\leq M\) and the \(a_ i\) are nonnegative integers less than p, with \(a_ L\) odd and all other \(a_ i\) even. Moreover, for each such n, there exists a semiprime ring of characteristic p having a derivation of nilpotency n.