I had to shorten the title because the whole thing wouldn't fit. I'll write the full question here and show my thoughts below.
For a commutative ring $R$, consider the power series ring $R[[x]]$ and the submonoid $S=\{x^n|n\in\mathbb N\}$ of $(R[[x]],\cdot)$ generated by $x$, let $R((x))$ be the localization $S^{-1}R[[x]]$. $R((x))$ is called a Laurent series ring over $R$, and its elements are called Laurent series.
Show that the elements of $R((x))$ may be written as $$\sum_{i=m}^\infty a_ix^i$$ where $m\in\mathbb Z$ and $a_i\in R$. (Note: $m$ represents a lower bound for the nonzero coefficients of a given series, but different series may have different values of $m$.)
I need a lot of help. When I'm not sure where to start, I usually just write down what I know...
I know that the power series ring $R[[x]]$ is the set of all polynomials like so $\{a_0, a_0 + a_1x, a_0 + a_1x + a_2x^2, \dots \}$. I know that $S$ is written out as $\{1, x, x^2, \dots \}$. I think I know the localization of $R[[x]]$ at $S$ is the "smallest" ring in which elements of $S$ become invertible. That is to say that $R((x))=S^{-1}R[[x]]$ is the smallest ring that makes $S$ invertible (do I have that right?). If that's the case, then I need the set of things that take elements of $S$ to the identity $(1,0,0, \dots)$.
So if I start with $n=0$, that's already the identity, so I'll need the identity $(1,0,\dots) \in R((x))$. If $n=1$ (that's $x$) so I'll need $x^{-1} \in R((x))$. For $x^k$, I will need $x^{-k} \in R((x))$. I haven't gotten to the part where I am able to write elements of $R((x))$ as how I want it, but am I on the right track in my initial thinking?
Thank you for any help.
I don’t think you’re thinking of elements of $R[[x]]$ in the most productive way. Just think of them as the kind of series you dealt with in Calculus, but not as functions, just as (usually) infinite expressions of the form $a_0+a_1x+a_2x^2+\cdots$. Make sure you understand, for example, why the element $1-x\in R[[x]]$ has for its reciprocal the element $1+x+x^2+x^3+\cdots\in R[[x]]$.
Now, when you make the single element $x$ invertible in the extension ring, you get things like $x^{-3} +2x^{-1}-1+3x^3-x^4$, just as an example of a series with finitely many terms; and you should persuade yourself that the reciprocal of $x^m(1-x)$ is $\sum_{n=-m}^\infty x^n$.
Once you are comfortable dealing with power series, the concept of Laurent series will come fairly easily to you.