INTEGRAL DOMAINS OF THE FORM A + XI[[X]]: PRIME SPECTRUM, KRULL DIMENSION

Abstract

Let A be an integral domain, X an analytic indeterminate over A and I a proper ideal (not necessarily prime) of A. In this paper, we study the ring [Formula: see text] First, we study the prime spectrum of R. We prove that the contraction map: Spec (A[[X]]) → Spec (R); Q ↦ Q ∩ R induces a homeomorphism, for the Zariski's topologies, from {Q ∈ Spec (A[[X]]) | XI[[X]] ⊈ Q} onto {P ∈ Spec (R) | XI[[X]] ⊈ P}. If P ∈ Spec (R) is such that XI[[X]] ⊆ P then there exists p ∈ Spec (A) such that P = p + XI[[X]]. Next, we study the Krull dimension of R. We give a necessary condition for R to be of finite Krull dimension. In particular, if R is of finite dimension then I must be an SFT ideal of A. Then we determine bounds for dim (R). Examples are given to indicate the sharpness of the results. In case I is a maximal ideal of A and A is either a Noetherian ring, SFT Prüfer domain or A[[X]] is catenarian and I SFT, we establish that dim (R) = dim (A[[X]]) = dim (A) + 1. Finally, we examine the possible transfer of the LFD property and the catenarity between the rings A, A[[X]] and R in case I is a maximal ideal of A.