Let $R$ be a commutative ring and $S$ be a multiplicative subset of $R$ and let $R[S^{-1}]$ be the localization of $R$ at $S$. I want to characterize the nilpotent elements in $R[S^{-1}]$, i.e. the elements $x \in R[S^{-1}]$ such that $\exists n \in \mathbb{N}$ with $x^n = 0$. Is it just the zero element?
My approach is this: let $ x \in R[S^{-1}]$ aka $ x = \frac{a}{b}, a \in R, b \in S$ and let $x$ be nilpotent, aka $(\frac{a}{b})^n = 0$ for some $n$. But then alls sorts of things can happen in such an abstract setting no?Does it follow that $a^n = 0$? We're not even in an Integral domain... And I know the denominator can have surprising behavior too in this type of construction...