ABSTRACT
Proof. Since a is surjective, cr(T) = T and it follows by induction that crl(T) = T for i > 1. Now consider the sequence
ker a C ker a2 C • • •
of ideals of T. Since T is Noetherian, the sequence stabilizes, say ker an+1 = ker a". Let x € kercr. Since an(T) = T, there exists an element y £ T such that <Jn(y) = x. Then an+1(y) = <r(z) = 0 and so y e ker<7™+1 = kera". Hence, 0 = un(y) = x and ker cr = 0. D
We next show that <p/ is a surjection if and only if it is an automorphism. The proof is adapted from [1], which was published four years after Gilmer's paper. Proposition 2. With the notation as above, iff is a surjection if and only if it is an automorphism.
