On nilpotency of the separating ideal of a derivation

Abstract

We prove that the separating ideal S ( D ) S(D) of any derivation D D on a commutative unital algebra B B is nilpotent if and only if S ( D ) ∩ ( ⋂ R n ) S(D) \cap (\bigcap {{R^n})} is a nil ideal, where R R is the Jacobson radical of B B . Also we show that any derivation D D on a commutative unital semiprime Banach algebra B B is continuous if and only if ⋂ ( S ( D ) ) n = { 0 } \bigcap {{{(S(D))}^n} = \{ 0\} } . Further we show that the set of all nilpotent elements of S ( D ) S(D) is equal to ⋂ ( S ( D ) ∩ P ) \bigcap {(S(D) \cap P)} , where the intersection runs over all nonclosed prime ideals of B B not containing S ( D ) S(D) . As a consequence, we show that if a commutative unital Banach algebra has only countably many nonclosed prime ideals then the separating ideal of a derivation is nilpotent.