[FOM] Jan Pax's question on Ring

smohan at isical.ac.in smohan at isical.ac.in
Fri Aug 2 01:46:23 EDT 2013

Response to Jan Pax's question on rings:

If I have understood the question correctly, the answer is no even if
$R$ is a field:

The field of rational functions $R(X)$ is neither contained in nor
contains the ring of power series $R[[X]]$.

It is clear that $X$ has no inverse in in $R[[X]]$. So, $1/X$ is in $R(X)$
(the field of rational functions in $X$ over $R$) but has no power series
expansion.

The exponential power series $\sum X^n/n!$ does not belong to any
extension field of $R$ of finite transcendence degree but $K$ is of
transcendence degree 1. So, the exponential power series does not belong
to $R(X)$.
''

SHASHI M. SRIVASTAVA