Commutative Ring Theory

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

M. FONTANA Dipartimento di Matematica, Terza Università degli Studi di Roma, 00146 Roma, Italy

L. IZELGUE and S. KABBAJ Département de Mathématiques et Informatique, Faculté des Sciences “Dhar El-Mehraz”, Université de Fès, Fès, Morocco

0. Introduction

All the rings considered in this paper are integral domains, i.e. commutative rings with identity and non zero-divisors. Given a finite dimensional ring A, we say that A is a Jaffard domain if dim A = dim_v A [2]. The previous property is not a local property and thus we say that A is a locally Jaffard domain if $A_{\mathfrak{P}}$ is a Jaffard domain, for each prime ideal $\mathfrak{P}$ of A. Noetherian domains and, in the locally finite dimensional case, Prüfer domains, ...