On the group actions of the units on non-units in a compact ring (Q1105676)

Let R be a ring with identity and G the group of its units. If H is a subgroup of G then \(R^ H=\{x\in R:\) \(hx=x\) for all \(h\in H\}\), \(I_ HR=\sum_{h\in H}(1-h)R\). If H is finite then \(N_ H: R\to R\), \(N_ H(r)=\sum_{h\in H}hr\), \(r\in R\) is a homomorphism of left H-modules. The authors study properties of the action of the group G of units of a topological ring R on the additive group of R. It is proved that for a compact semisimple ring R, \(R^ H= N_ HR\) for every finite subgroup H of G iff R is isomorphic to the ring of \(2\times 2\) matrices over \({\mathbb{Z}}/(2)\) or to the ring \(\prod_{\alpha \in \Omega}F_{\alpha}\), where each \(F_{\alpha}\) is a finite field and for \(\alpha\neq \beta\) \((char F_{\alpha},| F_{\beta}| -1)=1\). Moreover, it is proved that if R is a compact local ring, char R\(=p\) m, \(p\neq 2\), then \(R^ H\) and \(R/I_ HR\) are isomorphic as right R-modules for every subgroup H of G and \(R^ H=N_ HR\) for every finite subgroup H of G iff R is isomorphic to a finite field or to the ring \({\mathbb{Z}}/(p^ m).\)