A note on power invariant rings

Abstract

Let R be a commutative ring with identity and R((n))=R[[X1,…,Xn]] the power series ring in n independent indeterminates X1,…,Xn over R. R is called power invariant if whenever S is a ring such that R[[X1]]≅S[[X1]], then R≅S. R is said to be forever-power-invariant if S is a ring and n is any positive integer such that R((n))≅S((n)) then R≅S Let IC(R) denote the set of all a∈R such that there is R- homomorphism σ:R[[X]]→R with σ(X)=a. Then IC(R) is an ideal of R. It is shown that if IC(R) is nil, R is forever-power-invariant