The connection between the $\pi$-regularity of an associative ring with 
identity and the simplicity of all of its prime factors has long been 
investigated by many authors.  We prove, in this note, that for a ring 
$R$ whose maximal right (or left) ideals are two-sided, 
the following conditions are equivalent:
(1) Every prime ideal of $R$ is maximal; (2) Every prime ideal of $R$ is 
primitive; (3) Every prime factor of $R$ is Artinian; (4) Every prime factor 
of $R$ is von Neumann regular; (5) $R$ is $\pi$-regular and $R/P(R)$ is 
strongly regular; (6) $R$ is strongly $\pi$-regular and $R/P(R)$ is strongly 
regular; (7) $R/P(R)$ is strongly regular.  This generalizes 
and sharpens all the known results in ~\cite{Storrer-68}, 
~\cite{Chandran-77}, ~\cite{Hirano-78}, ~\cite{Yao-85} and ~\cite{Yu-94}.