On a partially ordered set associated to ring morphisms

We associate to any ring R with identity a partially ordered set Hom(R), whose elements are all pairs (a,M), where a=ker⁡φ and M=φ−1(U(S)) for some ring morphism φ of R into an arbitrary ring S. Here U(S) denotes the group of units of S. The assignment R↦Hom(R) turns out to be a contravariant functo...

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Bibliographic Details
Published in:Journal of algebra 2019-10, Vol.535, p.456-479
Main Authors: Facchini, Alberto, Heidari Zadeh, Leila
Format: Article
Language:English
Subjects:
Contravariant functor
Partially ordered set
Ring morphism
Universal inverting mapping of rings
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Summary:We associate to any ring R with identity a partially ordered set Hom(R), whose elements are all pairs (a,M), where a=ker⁡φ and M=φ−1(U(S)) for some ring morphism φ of R into an arbitrary ring S. Here U(S) denotes the group of units of S. The assignment R↦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 (a,M) in Hom(R) has a canonical representative, that is, there is a universal ring morphism ψ:R→S(R/a,M/a) corresponding to the pair (a,M), where the ring S(R/a,M/a) is constructed as a universal inverting R/a-ring in the sense of Cohn. Several properties of the sets Hom(R) and Max(R) are studied.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2019.06.027