A note on some chain conditions (Q2480701)

Let \(R\) be a ring. For a positive integer \(n\), denote by \(R[n]\) the ring of polynomials in \(n\) indeterminates with coefficients in \(R\). Recall from [\textit{O. Echi}, Port. Math. 50, No. 3, 277--295 (1993; Zbl 0828.13009)] that a domain \(R\) is \(E_n\) (resp. \(C_n\)) if, for each saturated chain \(P_0\subset P_1\subset P_2\) of prime ideals of \(R[n]\) such that \(P_0\cap R=P_1\cap R\), we have ht\((P_2/P_0)=2\) (resp. for each consecutive pair \(P'\subset P\) of prime ideals in \(R[n]\), letting \(p'=P'\cap R\), we have ht\((P/p'[n])=\)ht\((P'/p'[n])+1\)). An \(E\)-ring (resp. a \(C\)-ring) is a ring \(R\) which is \(E_n\) (resp. \(C_n\)) for each \(n\). It is known that for a ring \(R\), we have : \(R\) is \(E_n\) (resp. \(C_n)\Rightarrow R\) is \(E_{n-1}\) (resp. \(C_{n-1}\)); \(R\) is \(C_n\Rightarrow R\) is \(E_n\); and \(R\) is \(C_1\Leftrightarrow R\) is \(E_1\). In this short note, the author answers by the negative the following question raised by Echi in [loc. cit.]: Is an \(E_1\) (resp. a \(C_1\))-ring an \(E\) (resp. a \(C\))-ring? For a prime integer \(p\), setting \(R={\mathbb Z}+p{\mathbb Z}[[t]]\), the author shows that \(R\) is \(C_1\) (thus \(E_1\)) but \(R\) is not \(E_2\) (thus not \(C_2\)).