On weakly-Krull domains of integer-valued polynomials (Q6601232)

Let \(D\) be an integral domain with quotient field \(K\) and let \(\mathrm{Int}(D) := \{ f\in K [X]; f (D)\subseteq D\}\) be the ring of integer-valued polynomials over D. The paper under review deals with the question of when \(\mathrm{Int}(D)\) is a weakly-Krull domain. The main result asserts that \(\mathrm{Int}(D)\) may be weakly-Krull only in the trivial case, that is, \(\mathrm{Int}(D)\) is weakly-Krull if and only if \(\mathrm{Int}(D) = D[X]\) and \(D\) is a weakly-Krull \(UMT\)-domain. As an immediate consequences, for a Noetherian domain \(D\), \(\mathrm{Int}(D)\) is weakly-Krull if and only if \(D\) is weakly-Krull and \(\mathrm{Int}(D) = D[X]\). Also, for a domain \(D\), \(\mathrm{Int}(D)\) is an infra-Krull (resp. a generalized Krull) domain if and only if \(D\) is an infra-Krull (resp. a generalized Krull) domain and \(\mathrm{Int}(D)=D[X]\).