Two counterexamples about the Nagata and Serre conjecture rings

We construct two counterexamples to the open questions : is R n strong S(resp. catenary) when R(n) is ? The first example is a ring R such that R(n) is strong S and R n is not. The second is a stably strong S-domain R such that for all n≥1 and n=∞, R(n) is catenary and R n is not.