Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$

Question

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 $n\in\mathbb{N}$, $a^n\in R$.

Answer 1

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 .