Let $R$ be an integral domain with fraction field $K$, and let $I$ be an $R$-submodule of $K$. We say that $I$ is a fractional ideal of $R$ if $rI\subset R$ for some nonzero $r \in R$.
My question is: Is there any example of an $I$ which is not fractional? Please give as many examples as you can.
Your help (and hints) will be appreciated.
On
See this question: A question about the relationship between submodule and ideal. The goal of the introduction of fractional ideals is to extend the set of ideals of a ring by their "inverses". So every ideal is a fractional ideal but not inversely. As an example take $\theta=\sqrt{-5}$ and let the ideal $I_1=(3,1-\theta)$ and $I_2=(3,1+\theta)$ then the product of the two ideals $I_1I_2=(3)$, the principal ideal generated by $3$. Now we can say that $I2\over{3}$ is the inverse of the ideal $I_1$ but it is not an ideal since it contains elements that are not algebraic integers, but it is a fractional ideal because when multiplied with $3$ it becomes an ideal (whence the definition of fractional ideal).
Take $R=\mathbb Z$ so that $K=\mathbb Q$. Fix a finite set of primes $S$. Then $I_S=\{\frac{a}{b}\colon p\nmid b \,\,\forall p\in S\}$ is an $R$-module which is not a fractional ideal.