DERIVATIONS MAPPING INTO THE RADICAL .3.

We obtain commutativity-free characterizations of those derivations d on a unital complex Banach algebra A that map A into its radical: dA s ubset of or equal to rad(A) if and only if there exists a constant M g reater than or equal to 0 such that r(dx) less than or equal to Mr(x) for all x is an element of A, which in turn is equivalent to sup{r(z(- 1)dz)\z is an element of A invertible} < infinity (where r(.) is denot ing the spectral radius). The second characterization answers positive ly a question raised by J. Zemanek. (C) 1995 Academic Press, Inc.