ABSTRACT
In the last thirty years or so, a great deal has been learned about Mori domains, those integral domains that satisfy the Ascending Chain Condition on divisorial ideals. (A particularly good source for information is [2].) For one it is known that such domains can also be characterized as those that satisfy the Descending Chain Condition when the chains are restricted to being descending chains of divisorial ideals which have nonzero intersection [14, Theoreme I.I]. We shall consider both ACC and DCC on certain types of divisorial ideals in commutative rings with zero divisors. Of the two, ACC is the easier to state, we simply ask that the ring satisfy ACC on a particular type of divisorial ideal. For DCC we will always have some additional restriction placed on the intersection of the ideals in the descending
chain. When we say that a ring satisfies the restricted DCC on a certain type "X" of divisorial ideal, we will mean that the intersection of the ideals in the chain also is of type "X". For a Mori domain, we would simply say that the domain in question satisfies the restricted DCC on nonzero divisorial ideals, meaning that we only care what happens to a descending chain of divisorial ideals when the intersection of the ideals in the chain is known to be nonzero. We deal with two different types of divisorial ideals in this paper. Namely, regular divisorial ideals and semiregular divisorial ideals where "regular" means the ideal contains an element that is not a zero divisor (i.e., a regular element) and "semiregular" means the ideal contains a finitely generated ideal which has no nonzero annihilators.
