Maximal covers of chains of prime ideals

Original Paper
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Abstract

Suppose \(f:S \rightarrow R\) is a ring homomorphism, with S and R not necessarily commutative and f not necessarily unitary. We study the connections between chains in \(\text {Spec} (S)\) and \(\text {Spec} (R)\). We focus on the properties lying over, incomparability, going down (GD), going up (GU) and strong going between (SGB). We prove necessary and sufficient conditions for f to satisfy each of the properties GD, GU and SGB, in terms of maximal \(\mathcal D\)-chains, where \({\mathcal {D}} \subseteq \text {Spec} (S)\) is a nonempty chain. We show that if f satisfies all of the above properties, then every maximal \(\mathcal D\)-chain is a maximal 1:1 cover of \(\mathcal D\). We also present an important example from quasi-valuation theory in which all of the above properties are satisfied. Moreover, we give equivalent conditions for the following property: for every chain \(\mathcal D \subseteq \text {Spec} (S)\) and for every maximal \(\mathcal D\)-chain \(\mathcal C \subseteq \text {Spec} (R)\), \(\mathcal C\) and \(\mathcal D\) are of the same cardinality.

Keywords

Prime spectrum Going up Incomparability Going down Strong going between Lying over Maximal \({\mathcal {D}}\)-chains Maximal covers Layer n property 

Mathematics Subject Classification

16P70 13B21 16D25 
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Notes

Acknowledgements

The author would like to thank the referee for his/her careful reading and helpful comments.

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Authors and Affiliations

  1. 1.Sce CollegeAshdodIsrael

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