A ring A is called power
invariant if whenever B is a ring such that the formal power series rings A[[X]] and
B[[X]] are isomorphic, then A and B are isomorphic. A ring A is said to be strongly
power invariant if whenever B is a ring and ϕ is an isomorphism of A[[X]]
onto B[[X]], then there exists a B-automorphism ψ of B[[X]] such that
ψ(X) = ϕ(X). Strongly power invariant rings are power invariant. For any
commutative ring A,A∕J(A)n is strongly power invariant, where J(A) is the
Jacobson radical of A, and n is any positive integer. A left or right Artinian
ring is strongly power invariant. If A is a left or right Noetherian ring, then
A[t], the polynomial ring in an indeterminate t over A, is strongly power
invariant.
