Let $R,\Omega$ be two integral domains such that $R$ is Noetherian and $\Omega=R[\alpha]$ for some $\alpha\in \Omega$.


If there are infinite prime elements in $R$, can we proof that there are infinite prime elements in $\Omega$ too ?



A question about prime elements in a specific integral domain
An element $x \in A$ is prime if and only if the principal ideal $xA$ is prime.


Let the equation $y^2 = x^3 + Ax + B$ define an elliptic curve $E$ over an algebraically closed field $F$.


Let $R = F[x]$ and $\Omega = R[\sqrt{x^3 + Ax + B}]$, so that $\Omega$ is the coordinate ring of an affine patch of $E$.


Every nonzero prime ideal of $\Omega$ corresponds to a point on the elliptic curve, and thus has has nonzero image in the ideal class group (which is just the group of points, along with $\omega$, under the elliptic curve group law).


Consequently, $\Omega$ has no nonzero principal prime ideals at all, and thus no nonzero prime elements. However, $R$ has infinitely many prime elements, thus disproving your conjecture.


