**Journal of Prime Research in Mathematics**

Vol. 1 (2005), Issue 1, pp. 178 – 183

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

# 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

Copyright © 2005 Shaheen Nazir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**December, 2005.

### 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 \(R_{G}\).