Krull and Valuative Dimensions of the A + XB[X] Rings

All the rings considered are integral domains, i.e. commutative rings with identity and non zero-divisors. The previous property is not a local property and thus authors say that finite dimensional ring A is a locally Jaffard domain if AP is a Jaffard domain, for each prime ideal P of A. Noetherian domains and, in the locally finite dimensional case, Prufer domains, stably strong S-domains and universally catenarian domains are examples of locally Jaffard domains. As a matter of fact, the locally Jaffard domains coincide with the rings satisfying the inequality formula. Besides the locally Jaffard domains, further examples of Jaffard domains are given by the polynomial rings with the coefficients on a Jaffard domain and by some class of rings arising from the pullback diagrams of a special type. The chapter takes care of the theory of the dimension and of the transfer of the related properties in the constructions.