I don't understand the following:
Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ other than $\{0\}$, for which a $0\neq r\in R$ exists, so that $r I \subseteq R$ is an ideal of $R$.
A subset of $K$ defined as $(x) := R x$ with $x \in K^\times$ is called fractional principal ideal.
A fractional principal ideal $(x)$ is a fractional ideal because $x^{-1} (x) = R$
Well, but isn't it the case that $x^{-1}$ might not be an element of $R$!!?
Personally I would explain it that way:
Since $x\in K$, we have $x = \frac{a}{b}$ with $a, b \in R$ and $b \neq 0$, so $b (x) = b R x = \{ b r x : r \in R\} = \{ r a : r \in R\}$ which is an ideal in $R$, therefore $(x)$ must be a fractional ideal.
Is that ok?