Communications of the
Korean Mathematical Society CKMS

We show that a strong system of derivations $\{D_0 , D_1,,$ \par \noindent $\cdots D_m \}$ on a commutative Banach algebra $A$ is contained in the radical of $A$ if it satisfies one of the following conditions for separating spaces ; \roster \item"{(1)}" $\frak S (D_i ) \subseteq rad(A)$ and $\frak S(D_i ) \subseteq K_{D_i } (rad(A))$ for all $i$, where $$ K_{D_i } (rad(A)) = \{ x \in rad(A) : \text {for each }\ m \ge 1, \ D_i^m (x) \in rad(A) \}. $$ \item"{(2)}" $\frak S(D_i^m ) \subseteq rad(A)$ for all $i$ and $m$. \item"{(3)}" $\overline {x \frak S(D_i )} = \frak S(D_i )$ for all $i$ and all nonzero $x$ in $rad(A)$. \endroster

Keywords: derivation, higher derivation, Banach algebra

MSC numbers: 46H99