If $R$ is a unique factorization domain and $F$ its field of fractions then any $z\in F$ can be represented as an irreducible fraction (i.e. $z=a/b$ with $a,b\in R$ so that $a$ and $b$ have no common non-unit factors in $R$).
What is an example of a domain $R$ where this fails, i.e., a domain $R$ and an element $z$ in the field of fractions of $R$ so that $z$ has no representation as an irreducible fraction (meaning $z=a/b$ with $a,b\in R$ always implies that $a$ and $b$ have a common non-unit factor in $R$)?
Let $R=\mathbb Z+(X,Y)\mathbb Q[X,Y]$, and $z=\dfrac XY$.
It's obvious that every non-zero integer $n$ divides $X$ and $Y$: we have $X=n\dfrac Xn$ and $Y=n\dfrac Yn$.
Now write $z=\dfrac ab$ with $a,b\in R$, $b\ne0$. It's easily seen that $a(0,0)=0$, and $b(0,0)=0$. This shows that $a,b\in(X,Y)\mathbb Q[X,Y]$, and once again every non-zero integer $n$ divides $a$ and $b$. (In fact, $a=Xf$ and $b=Yf$ for some $f\in\mathbb Q[X,Y]$.)