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- W2896674166 abstract "We associate to any ring $R$ with identity a partially ordered set Hom$(R)$, whose elements are all pairs $(mathfrak a,M)$, where $mathfrak a=kervarphi$ 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 assignment $Rmapsto{}$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 $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." @default.
- W2896674166 created "2018-10-26" @default.
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- W2896674166 date "2018-10-14" @default.
- W2896674166 modified "2023-09-27" @default.
- W2896674166 title "On a partially ordered set associated to ring morphisms." @default.
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