Lifting chains of prime ideals to paravaluation rings
Let \(R\subset T\) be a unital extension of commutative rings and let \(\{P_\alpha \mid\alpha \in I\}\) be a chain of prime ideals of \(R\). It is proved that \(R\) is contained in some paravaluation ring \(V\) of \(T\) such that \(V\) has a chain of prime ideals \(Q_\alpha\) satisfying \(Q_\alpha\cap R=P_\alpha\) for each \(\alpha\in I\). It can be further arranged that \(N\cap R=\bigcup P_\alpha\) where \(N\) is such that \((V,N)\) is a paravaluation pair of \(T\). This generalizes a similar result by \textit{Byung Gyun Kang} and \textit{Dong Yeol Oh} [proved in Proc. Am. Math. Soc. 126, No. 3, 645-646 (1998; Zbl 0887.13004)] when \(T\) is a field. The paper contains a sufficient condition for deleting the prefix `para' from the above result and also contains some interesting open problems.
scientific article; zbMATH DE number 1462348