On a partially ordered set associated to ring morphisms

Abstract

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.