What does $R[[X]]$ and $R(X)$ stands for?

Question

I'm reviewing Linear Algebra these days and I saw these two notations in my notes without definition.

Those are, $R[[X]]$ and $R(X)$ where $R$ is a commutative ring with unity.

I remember that one of these denote the field of polynomials, but I don't know which one does..

Moreover, is there any notation for the set of polynomial functions?

Hoffman&Kunze's text denote it as $R[X]^{\sim}$ btw.

Answer 1

Typically:

• $R[x]$ denotes the set of polynomials over $R$
• When $R$ is a domain, $R(x)$ denotes the set of rational polynomials over $R$
• $R[[x]]$ denotes the formal power series over $R$
• $R((x))$ denotes the Laurent series over $R$

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vuur asked an interesting question in the comments which I can speak to here. The answer is "If $R$ is a commutative domain, then yes, $R(x)$ is the field of fractions for $R[x]$, and $R((x))$ is the field of fractions for $R[[x]]$. In that case, $R((x))$ can be expressed as "quotients of power series."

What's going on here is that $R(x)$ is almost always defined as quotients of polynomials, and that necessitates $R$ (and hence $R[x]$) to be at least a domain, so that the product of two denominators is nonzero. However, $R((x))$ is not usually defined via quotients, it's usually described as "power series, but you can have negative powers of $x$, and you can start at any power of $x$ and go upward." Thus $R((x))$ is defined for any ring $R$, but it is not necessarily quotients of things from $R[[x]]$.