  We associate to any ring $R$ with identity a partially ordered set Hom$(R)$,
whose elements are all pairs $(\mathfrak a,M)$, where $\mathfrak a=\ker\varphi$
and $M=\varphi^{-1}(U(S))$ for some ring morphism $\varphi$ of $R$ into an
arbitrary ring $S$. Here $U(S)$ denotes the group of units of $S$. The
assignment $R\mapsto{}$Hom$(R)$ turns out to be a contravariant functor of the
category Ring of associative rings with identity to the category ParOrd of
partially ordered sets. The maximal elements of Hom$(R)$ constitute a subset
Max$(R)$ which, for commutative rings $R$, can be identified with the Zariski
spectrum Spec$(R)$ of $R$. Every pair $(\mathfrak a,M)$ in Hom$(R)$ has a
canonical representative, that is, there is a universal ring morphism
$\psi\colon R\to S_{(R/\mathfrak a,M/\mathfrak a)} $ corresponding to the pair
$(\mathfrak a,M)$, where the ring $S_{(R/\mathfrak a,M/\mathfrak a)} $ is
constructed as a universal inverting $R/\mathfrak a$-ring in the sense of Cohn.
Several properties of the sets Hom$(R)$ and Max$(R)$ are studied.
