# Extended ideals in power series ring

Let $A$ be a commutative ring with $1$ and consider the ring of formal power series $A[[X]]$. If $I \subseteq A$ is an ideal, let $I[[X]]$ denote the set of power series with coefficients in $I$. This is an ideal; it is the kernel of the reduction homomorphism $A[[X]] \to (A/I)[[X]]$. Let $IA[[X]]$ denote the extended ideal.

Matsumura writes the following in his Commutative Ring Theory (p. 5):

...if $I$ is not finitely generated then $I[[X]]$ is bigger than $IA[[X]]$.

I believe if $I$ has a countably infinite set of generators, say $I=(a_0,a_1,\ldots)$, then the following will suffice as a proof: If $f=a_0 + a_1 X + a_2 X^2 + \cdots$, then $f$ is not in the exteneded ideal. Indeed, if it were, then $f = \sum_{i=1}^m b_i g_i$ for some $b_i \in I$ and $g_i \in A[[X]]$. But then $(a_0,a_1,\ldots) \subseteq (b_1,b_2,\ldots,b_m)$, so $I$ is finitely generated.

What is the proof for a general non-countably-generated ideal? My ideas aren't leading anywhere.

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I think Matsumura meant "if $I$ is not finitely generated, then usually $I[[X]]$ is bigger than $I A[[X]]$". – Martin Brandenburg Jun 10 '13 at 14:47
@MartinBrandenburg, I was suspicious that was what he meant, but I wasn't sure. Thanks. – John Myers Jun 10 '13 at 15:35
@YACP, It's verbatim from Matsumura. I wrote the dot because I wanted it there. – John Myers Jun 19 '13 at 15:06

$I[[X]]=I A[[X]]$ holds if and only if for every at most countable subset $S \subseteq I$ there is some finite subset $B \subseteq I$ such that $(S) \subseteq (B)$. Clearly this holds when $I$ is finitely generated. But the converse is not true. There are examples with $|I|=\aleph_1$, the smallest uncountable cardinality. I think $A=k[t,\{X_{\alpha}\}_{\alpha<\aleph_1}]/(X_\alpha=t X_{\beta})_{\alpha<\beta}$ and $I=(X_{\alpha})_{\alpha<\aleph_1}$ should work.
Maybe I have misunderstood something but it seems that $I$ is a principal ideal generated by $X_0$ since $X_\alpha=tX_{\alpha+1}=X_0$ for any $\alpha$. – curious Feb 22 '15 at 22:41
You are right. What abuut the quotient by $(X_{\alpha} = t X_{\alpha+1})$? – Martin Brandenburg Mar 2 '15 at 1:04