On a partially ordered set associated to ring morphisms

TL;DR

This paper introduces a new partially ordered set associated with any ring, linking ring morphisms to order theory, and explores its properties and relation to the Zariski spectrum for commutative rings.

Contribution

It defines a novel partially ordered set Hom(R) for rings, establishes its functorial properties, and relates maximal elements to the Zariski spectrum in the commutative case.

Findings

Hom(R) is a contravariant functor from rings to partially ordered sets.

Max(R) corresponds to the Zariski spectrum for commutative rings.

Universal morphisms are constructed for each element of Hom(R).

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.