On the Module Structure of the Integer-Valued Polynomial Rings

The ring Int (D) : = { f∈ K[X] : f(D) ⊆ D} , where D is an integral domain with quotient field K, is known to be a D-module. We present some progress in the classification of integral domains D such that Int (D) is locally free, or flat, over D. Particularly, we are interested in locally essential domains. We also investigate the question of whether the flatness of Int (D) as a D[X]-module forces Int (D) = D[X] in various contexts of integral domains such as domains that are: essential, strong Mori, GCD. Interesting results are established with some applications and illustrating examples.