About the Spectrum of the RingsR(n) andR⟨n⟩

The purpose of this paper is to find necessary and sufficient conditions for the transfer of the strong S, catenarian, residually Jaffard and totally Jaffard properties between the rings R(1) and R⟨1⟩ for domains R of dimension ≤ 2. Moreover, we give a positive answer to a conjecture of S. Kabbaj in the case of 2-dimensional integrally closed strong S-domains; we prove that if R is a 2-dimensional integrally closed strong S-domain then R[1] is catenarian.