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Normally, one defines formal power series as below:

Let $F$ be a field. A formal power series is an expression of the form $$ a_0 + a_1x + a_2x^2 + \dots = \sum_{n \geqslant 0} a_nx^n,$$ where $\{a_n\}_{n \geqslant 0} \subset F$ is our sequence of coefficients.

My question. Can we extend (somehow) this definition by taking rings instead of fields? I.e., can we define a formal power series as an expression of the form

$$ b_0 + b_1x + b_2x^2 + \dots = \sum_{n \geqslant 0} b_nx^n, $$ where $\{b_n\}_{n \geqslant 0} \subset R$ ($R$ is an arbitrary ring) is our sequence of coefficients?

Motivation of the question. This question came up to my mind based on this two questions and their respective comments:

question 1 ; question 2

In this questions, we treat Legendre polynomials (which belong to $\mathbb R[x]$)as a sequence of coefficients for our formal power series. Certainly, we can't guarantee that $\mathbb R[x]$ is a field, and neither can we guarantee that the sequence of Legendre polynomials form a field, since $x = L_1(x)$ has no inverse. Therefore, the question I wrote above urged to my brain.

Thanks for any help in advance.

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Yes, you can define $R[[x]]$ for any ring $R$, as is done on the wikipedia page:

https://en.wikipedia.org/wiki/Formal_power_series#

The special property of formal power series when $R=k$ is a field is that $k[[x]]$ becomes a complete discrete valuation ring. We define the order $v(f)$ to be $n_0$ where

$$f(x)=\sum_{n=n_0}^\infty a_nx^n$$ with $a_{n_0}\ne 0$ then $v:k[[x]]\to \mathbb{Z}$ satisfies

  1. $v(xy)=v(x)+v(y)$
  2. $v(x+y)\ge \operatorname{min}(v(x),v(y))$

$k[[x]]$ is a principal ideal domain, and has a unique maximal ideal $(x)$. A power series $f\in k[[x]]$ has a multiplicative inverse iff $v(f)=0$ iff $f\not \in (x)$ iff $f=\sum_{n=0}^\infty a_nx^n$ and $a_0\ne 0$.

As to the part about Legendre polynomials - it doesn't make sense to ask if `the sequence of Legendre polynomials form a field' because they don't form a ring. They are of course a basis for $\mathbb{R}[x]$, which is a ring, but like all polynomial rings it is not a field.

If you wanted to consider the ring of formal power series $R[[y]]$ where the coefficients $R=\mathbb{R}[x]$ are spanned by the Legendre polynomials, then that is perfectly well-defined.

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