  Let $\ast $ be a star operation of finite character. Call a $\ast $-ideal $I$
of finite type a $\ast $-homogeneous ideal if $I$ is contained in a unique
maximal $\ast $-ideal $M=M(I).$ A maximal $\ast $-ideal that contains a $\ast
$-homogeneous ideal is called $\ast $-potent and the same name bears a domain
all of whose maximal $\ast $-ideals are $\ast $-potent. One among the various
aims of this article is to indicate what makes a $\ast $-ideal of finite type a
$\ast $-homogeneous ideal, where and how we can find one, what they can do and
how this notion came to be. We also prove some results of current interest in
ring theory using some ideas from this author's joint work in \cite{LYZ 2014}
on partially ordered monoids. For example we characterize when a commutative
Riesz monoid generates a Riesz group.
