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.