Talk:Restricted power series

Confusing

@D.Lazard: I am bit confused by the confusing template. The reason given was “The article is mainly about series over a field, and the definition implies the existence of infinitely many ideals is the field”. I agree the most familiar case is, it seems, one over a non-archimedean field. But why is that confusing? My thinking was the article like this need not be too elementary; e.g., we can assume readers know projective limits for instance. Also, why is the number of ideals matter? (I feel like I’m missing something). Note if A is a discrete ring, the ring of restricted power is the same as a polynomial ring (so it’s generally not a field). —- Taku (talk) 00:01, 27 April 2020 (UTC)[reply]

In the definition of

A\langle x_{1},\dots ,x_{n}\rangle ,

you use a fundamental system of open ideals of

A

. Then you consider

k\langle x_{1},\dots ,x_{n}\rangle ,

where

k

is a field. This implies that

k

must be a linearly topologized ring, which is not possible, as a field does not have non-trivial ideals. I know that a way for solving the problem is to define a linearly topologized field as the field of fractions of a linearly topologized ring, but I am not enough aware of the subject for knowing whether it is the standard definition. In any case, the definition is not given here, nor in the linked article. D.Lazard (talk) 08:11, 27 April 2020 (UTC)[reply]

Yes, thank you for catching the error; I meant to have used the valuation ring. I have also added a reference to the definition (this isn’t really original research). —- Taku (talk) 16:23, 27 April 2020 (UTC)[reply]

I think somehow more official way to do is to define the module of restricted power series ring with coefficients in a linearly topologized module (we view the field of fractions as a module over the base domain). But, for this, it seems one needs the module to be finitely generated. —- Taku (talk) 16:40, 27 April 2020 (UTC)[reply]

I removed the confusing template for now, as I think the definition is now correct. -- Taku (talk) 00:09, 3 May 2020 (UTC)[reply]