Journal of Prime Research in Mathematics

# Continuous automorphisms and an equivalence relation In $$K[[X]]$$

Shaheen Nazir
School of Mathematical Sciences, Government College University, Lahore, Pakistan.

$$^{1}$$Corresponding Author: snazir@sms.edu.pk

### Abstract

Let K be an arbitrary commutative field and let $$R = K[[X]]$$ be the ring of formal power series in one variable. Let $$G_{R}$$ be the set of all power series of the form $$u = Xv$$, where $$v$$ is a unity in $$R$$. Relative to the usual composition $$G_{R}$$ becomes a topological group with respect to the $$X-$$adic topology of $$R$$. We also study an equivalence relation on $$R$$. Let $$R = K[[X]]$$ be the ring of formal power series in one variable over a fixed commutative field $$K$$. We denote by $$ord f = min \{i: a_{i} ≠ 0\}$$ for any $$f ∈ R$$ . It is well known that $$ord f$$ is a valuation on $$R$$ and $$R$$ becomes a complete topological ring relative to the topology induced by this valuation. Let  $$G_{G}=\{u∈ R: ord u=1\}$$ and for $$u,v ∈ R_{G}$$ we denote $$(uov)(X)=v(u(X))$$, a new element of $$R_{G}$$.