On the Jacobson radical of constants of derivations

Let R be a semiprimitive algebra, d   its algebraic derivation and Rd=kerd the subalgebra of constants of d  . It is proved that the Jacobson radical J(Rd)J(Rd) of RdRd is nilpotent. It is also shown that the following properties are equivalent: RdRd is semilocal; R   is semisimple Artinian; RdRd is left and right Artinian.