# [FOM] Ring Theory/Question

Thomas Kucera Thomas.Kucera at umanitoba.ca
Thu Aug 1 12:34:00 EDT 2013

The question really only makes complete sense when R is a (commutative) field; more things need to be specified in any more general settings.

In this case, all polynomials with non-zero constant term in R[\lambda] are invertible in R[[lambda]] (just use the ordinary algorithm for long division). So the usual ring-theoretic localization of R[\lambda] at the ideal generated by \lambda embeds in both R[[lambda]] and K; and in fact is the intersection of these two rings. If P is a Laurent series, then \lambda^n P is in R[[\lambda]] for some n, and so R((\lambda)) is generated as a ring by R[[\lambda]] and \lambda^{-1}, that is, by its subrings R[[\lambda]] and K.

---TK

On 2013-07-29, at 07:56 , pax0 at seznam.cz wrote:

> Dear All,
> let R be a ring.
> It is obvious that the following inclusions hold
>
> R ⊆ R[lambda] ⊆ R[[lambda]] ⊆ R((lambda)).
>
> Here, R[lambda] is the ring of polynomials in the variable lambda,
> R[[lambda]] is the ring of formal power series over R,
> and R((lambda)) is the ring of Laurent series.
>
> Q. ***Can I put in general the quotient field K of R[lambda] somewhere in this chain?***
>
> Any justification is welcome.
> Jan Pax
>
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