# Set of non-units in F[[x]]

• Aug 14th 2010, 10:12 PM
i_never_noticed
Set of non-units in F[[x]]
Hey,

Let F be any field and let I = F[[x]] \ F[[x]]* be the set of non-units in F[[x]].
Show that I is an ideal of F[[x]].

this question is troubling me as i cannot seem to properly define what I is, can someone please explain I to me?

Cheers
• Aug 15th 2010, 11:11 AM
NonCommAlg
easy to prove: $I=\{\sum_{n=0}^{\infty} a_nx^n \in F[[x]]: \ a_0=0\}= \langle x \rangle.$ in fact $I$ is the unique maximal ideal of $F[[x]]$ because $F[[x]]/I \cong F,$ which is a field.