On Mori domains and commutative rings with CC^⊥ I

We prove here, among other results, that if A is a Mori domain containing an uncountable field, then any polynomial ring over A is Mori. The Mori property of a domain is determined by its maximal ideals. We use extensively a known characterization of Mori domains in terms of the chain condition on annihilators. We characterize Mori domains in terms of semivaluations and divisibility groups. We prove an analogue of Nagata's theorem for the Mori property and present an example of a completely integrally closed domain A, S a multiplicative subset of A generated by prime elements, such that A is completely integrally closed, but A_S is not.