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If $R$ is a ring and $x$ is an indeterminate, what does the notation $R(x)$ mean? I've seen $R[x]$ but not $R(x)$.

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$R(x)$ usually means the field of rational functions over $R$, i.e $\{\frac{f}{g}: f \in R[x], 0 \neq g \in R[x]\}$. (Alternatively, you can view this as the field of fractions of $R[x]$).

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  • $\begingroup$ OK GREAT THANK YOU VERY MUCH $\endgroup$
    – HAHA
    Apr 8, 2015 at 15:19
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    $\begingroup$ Note that $R$ needs to be an integral domain for this to make sense. $\endgroup$
    – Seth
    Apr 8, 2015 at 15:20
  • $\begingroup$ OTHERWISE YOU CANNOT TAKE THE FIELD OF FRACTIONS, CORRECT? $\endgroup$
    – HAHA
    Apr 8, 2015 at 15:21
  • $\begingroup$ Yes. Please stop using all caps though. $\endgroup$
    – Seth
    Apr 8, 2015 at 15:22
  • $\begingroup$ SIR MY KEYBOARD IS BROKEN $\endgroup$
    – HAHA
    Apr 8, 2015 at 15:23

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