ring morphisms and their orderings

We associate to any ring $r$ with identity a partially ordered set $hom(r)$, whose elements are all pairs $(mathfrak a,m)$, where $mathfrak a=kervarphi$ 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 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 $psicolon rto 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.