## Mori domains.(English)Zbl 1079.13509

Chapman, Scott T. (ed.) et al., Non-Noetherian commutative ring theory. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-6492-9/hbk). Math. Appl., Dordr. 520, 57-73 (2000).
This paper reports on Mori domains. We cite from the author’s introduction: “A domain satisfying the a.c.c. on integral divisorial ideals is called a Mori domain. Krull domains as well as Noetherian domains are both Mori. However there exist large classes of Mori, non-Noetherian and non-Krull domains. – Two main questions arise in the study of Mori domains:
(1) Is the polynomial ring (and the power series ring) over a Mori domain still a Mori domain?
(2) Is the complete integral closure $$A^*$$ of a Mori domain $$A$$ a Krull domain?
Although the answers to the questions above are positive for Noetherian and for Krull domains, both of them turned out to have negative answers in case of Mori domains. Positive answers need some other hypotheses.”
The paper under review surveys the known results in connection with the questions above, especially in the cases when the Mori property is inherited by ring extensions and spezializations.
For the entire collection see [Zbl 0964.00012].

### MSC:

 13G05 Integral domains 13E99 Chain conditions, finiteness conditions in commutative ring theory