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>>632
> >>631 ついで
> 
> http://math.stackexchange.com/questions/271536/ring-of-formal-power-series-finitely-generated-as-algebra
> Ring of formal power series finitely generated as algebra?  asked Jan 6 '13 at 13:44 user55354
> 
> I'm asked if the ring of formal power series is finitely generated as a K-algebra. Intuition says no, but I don't know where to start. Any hint or suggestion?
> 
> 2 Answers
> 
> Let A be a non-trivial commutative ring. Then A[[x]] is not finitely generated as a A-algebra.
> 
> Indeed, observe that A must have a maximal ideal m, so we have a field k=A/m, and if k[[x]] is not finitely-generated as a k-algebra, then A[[x]] cannot be finitely-generated as an A-algebra. So it suffices to prove that k[[x]] is not finitely generated.
> Now, it is a straightforward matter to show that the polynomial ring k[x1,…,xn] has a countably infinite basis as a k-vector space, so any finitely-generated k-algebra must have an at most countable basis as a k -vector space.
> 
> However, k[[x]] has an uncountable basis as a k-vector space. Observe that k[[x]] is obviously isomorphic to kN, the space of all N-indexed sequences of elements of k, as k-vector spaces. But it is well-known that kN is of uncountable dimension: see here, for example.

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