# Does it make sense to think about formal power series where the coefficients belong to a Ring?

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.

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.