# The image of a derivation is contained in the radical

### Abstract

In 1955 I. M. Singer and J. Wermer proved that a bounded derivation on a commutative Banach algebra maps into the (Jacobson) radical; they conejctured that this result holds even if the derivation is unbounded. We give a proof of this conjecture. The central idea in the proof is the introduction of the concept of a recalcitrant system of elements in a commutative radical Banach algebra. Such systems put algebraic constraints upon a derivation which prevent the derivation from mapping outside of the radical.

Marc P. Thomas