The profinite structure of the group of units of a power series ring with coefficients in a finite ring. (Q2576975)

Let \(A\) be a finite unitary commutative ring, and \(A[\![T]\!]\) be the power series ring over \(A\). The group of units of \(A[\![T]\!]\) has the form \(A^*\times{\mathbf P}(A)\) where \(A^*\) is the group of units of \(A\) and \({\mathbf P}(A):=\{f\in A[\![T]\!]\mid f(0)=1\}\). The authors show that \({\mathbf P}(A)\) is a profinite group with the structure \(\Gamma\times\prod\mathbb{Z}_p^{\aleph_0}\) where \(\Gamma\) is a profinite Abelian group of finite exponent and the product in the second factor ranges over the primes \(p\) which divide \(|A|\). Suppose, furthermore, that \(A\) satisfies the condition (*): whenever \(p^\alpha a=0\) for some prime \(p\), \(\alpha\geq 1\) and \(a\in A\), we have \(p^{\alpha-1}a^p=0\). If (*) holds then \(\Gamma\cong N^{\aleph_0}\) where \(N\) is the nilradical of \(A\).