Suppose $R$ is integral domain and $K$ is the fraction field of $R$. Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$.
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I think you mean $n \in \mathbb{Z}, n\geq 2.$ otherwise it would be very trivial. for example take $R = \mathbb{Z}, a = \frac{1}{2}, n = -1$ – Krish Nov 29 '14 at 18:45
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Excuse me $n \in \mathbb(N)$. – m a s Nov 29 '14 at 18:50
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a) $R=\mathbb Q[T^2, T^3]\subset K=\mathbb Q(T), a=T, n=2$.
b) $R=\mathbb Z[2i]\subset \mathbb Q(i), a=i, n=2$
c) etc $\cdots$
I think you get the idea. The key concept here is (non-) integrally closed .
Georges Elencwajg
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Or better say non root closed. (In commutative algebra there is also the concept of root closure of an integral domain.) – user26857 Nov 29 '14 at 19:02
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@user26857: well, yes, but "root closed" is not a very widespread concept in commutative algebra or algebraic geometry... – Georges Elencwajg Nov 29 '14 at 19:06