Let $D$ be a UFD over a field $k$ of characteristic zero. Assume that $w$ is algebraic over $D$. Denote $R=D[w]$.
Observe that $R$ is not necessarily a UFD.
Can one find an example in which $R$ has no prime elements at all?
My example: $D=k[x^2]$, $w=x^3$ (its minimal polynomial over $k[x^2]$ is of degree $2$. Notice that $x^3$ is integral over $D$), $R=k[x^2][x^3]$; is it true that $k[x^2,x^3]$ has no prime elements? Of course, as a Noetherian ring (or more generally, as a ring which satisfies ACCP), $R$ has irreducible elements, for example, $x^2$ and $x^3$, but these are not prime elements since $(x^2)(x^2)(x^2)=(x^3)(x^3)$.
See also this question.
Remark: If $w$ is transcendental over $D$, then $R=D[w]$ is a UFD, as a polynomial ring (in one variable) over a UFD.