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Thread: Set of non-units in F[[x]]

  1. August 14th 2010, 10:12 PM #1
    i_never_noticed
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    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
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  2. August 15th 2010, 11:11 AM #2
    NonCommAlg
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    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.
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