On the dimension theory of polynomial rings over pullbacks

Since Seidenberg's (1953-54) papers [35, 36] and Jaffard's (1960) pamphlet [28] on the dimension theory of commutative rings, the literature abounds in works exploring the prime ideal structure of polynomial rings, including four pioneering articles by Arnold and Gilmer on dimension sequences [3, 4, 5, 6]. Of particular interest is Bastida-Gilmer's (1973) precursory article [8] which established a formula for the KruU dimension of a polynomial ring over a D + M issued from a valuation domain. During the last three decades, numerous papers provided in-depth treatments of dimension theory and other related notions (such as the S-property, strong S-property, and catenarity) in polynomial rings over various puUback constructions. All rings considered in this paper are assumed to be integral domains.