In the paper, we propose the definition of $U$-prime ideals to describe a class of positive-dimensional ideals. We study the properties of a $U$-prime ideal and prove that an ideal $I$ is $U$-prime if and only if $I=I^{\text{ec}}$. To decompose a $U$-prime ideal $I$ with $U$ maximally algebraically independent modulo $I$, we first show that the primary decomposition, radical of a zero-dimensional ideal and the prime decomposition of the radical of the ideal can be directly obtained from the Rational Univariate Representation (RUR) of it, then use the RUR of the extension ideal of a $U$-prime ideal $I$ to simultaneously compute the primary decomposition, radical of $I$ and the prime decomposition of $\sqrt{I}$. At last, we present a method to solve SHEPWM problems using $U$-primeness, which can give a ``nearly parametric" solution to a SHEPWM system.