Krull型整环与几类特殊整环

Krull type domains and other special domains

设R是有单位元的整环.本文用通常的星型算子来刻画Krull型整环与其它几类特殊整环之间的关系.本文证明了若dim(R)≥2,则R的每个素w-理想的高度为1当且仅当任给R的素理想P,若htP≥2,那么P是强w-可逆理想.另外,若R是Krull型整环,dim(R)≥2,w-dim(R)=1,且为H整环,那么,对任给R的素w-理想M,则M是w-可逆理想,当且仅当M不是强w-理想,当且仅当RM是离散赋值环,当且仅当RM是赋值环.同时,我们给出了有限特征的GCD整环与Krull型整环的一些等价条件.最后,我们论证了若R是Prufer整环,又是Krull型整环,任给非零非单位a∈R,则有R是阿基米德整环当且仅当a含在R的某个极小素理想中.

In this paper, we characterize Krull type domains by using general star operations. We characterize the relationships between Krull type domains and other special domains. Let R be a domain with dim（R）〉 2, we indicate that each prime ideal P with htP _〉 2 of R is a strongly W-invertible ideal, if and only if each prime W-ideal of R has the height of 1. Besides, we show that R is a Krull type domain and an H domain with dim（R）≥ 2 and W - dim（R）=1, then each prime W -ideal M of R is W-invertible, if and only ifM is not strong W-ideal, if and only if RM is a valuation domain, if and only if RM is a DVR. Moreover, we obtain the equivalent conditions between Krull type domains and GCD domains. Finally, we prove that a Krull type domain R is a Prufer domain, then R is an Archimedean domain if and only if α is contained in some minimal prime ideal of R for every nonzero nonunit a.