Remarks on quasinormal rings
Remarks on quasinormal rings
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关于拟正规环的评论
DOI：
10.1016/0022-4049(84)90026-4
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发表时间：
1984
影响因子：
0.8
通讯作者：
KEN
中科院分区：
文献类型：
--
作者：
Nobuharu Onoda;KEN
设A是整环，X是不定式.如果自然映射Pit A + Pic Al [X，Xl]是同构，我们称A是拟正规的.在H. Bass-MP Murthy [l]，S. Greco（31和DE Rush）[6]。本文的目的是补充拟正规环的一些新结果。环的拟正规性与环的拟正规性密切相关，这已经得到了很好的研究，例如，121，171，[8]。特别地，如果A是拟正规的，则A是拟正规的。而且。d1是正规的当且仅当A是（2，3）-闭的（cf.（二十一）、因此，很自然地要问是否存在这样一个准正规性的封闭型标准。t. Asanuma引入了tr-closedness的概念（参见。1.8)证明了：若dim A = 1，A是拟正规的当且仅当A是拟正规的且u-闭的.第一节的目的是寻找拟正规性与u-闭性之间的关系。我们的结果在引理1.10和推论1.11中给出。作为这些结果的结果，我们证明了Asanuma的上述定理（cf. 1.13 1.14）。在研究环的拟正规性时，一个障碍是拟正规性不是一个“整体到局部”的性质。也就是说，即使A是拟正规的，对于A的素理想p，A1不一定是拟正规的，然而，它是一个“局部到整体”的性质。也就是说，如果A对A的每一个素元p都是拟正规的，则A是拟正规的。如果A满足最后一个条件，则称环A是局部拟正规的。在第二节中，我们给出了A是局部拟正规的一些条件。主要结果是定理2.12。作为推论，我们证明了A是局部拟正规的当且仅当A是局部拟正规的且局部或闭的（参见。2.13）。第一作者对T. Asanuma指出了u-闭性在研究拟正规环中的重要性。
Let A be an integral domain and let X be an indeterminate. We call A quasinormal if the natural map Pit A+ Pic Al [X, Xl] is an isomorphism. The quasinormality of rings was studied in H. Bass-MP Murthy [l], S. Greco (31 and DE Rush [6]. The purpose of the present article is to add some new results on quasinormal rings as remarks to the above works. The quasinormality of rings is closely related to the seminormality of rings, which has been well studied, eg, 121, 171,[8]. In particular, A is seminoirmal if A is quasinormal. Moreover. dl is seminormal if and only if A is (2, 3)-closed (cf.(21). It is thus natural to ask whlellher or not such a closedness-type criterion of quasinormality exists. T. Asanuma introduced the notion of tr-closedness (cf. 1.8) and proved that, if dim A= 1, A is quasinormal if and only if A is seminormal and u-closed. The purpose of the first section is to look for a relationship between the quasinormality and the u-closedness. Our results are given in Lemma 1.10 and the Corollary 1.11. As a consequence of these results, we prove the above theorem of Asanuma (cf. 1.13 and 1.14). In studying the quasinormality of rings, an obstruction is that the quasinormality is not a ‘global-to-local’property. Namely, even if A is quasinormal, A, is not necessarily quasinormal for a prime ideal p of A, However, it is a ‘local-to-global’property. Namely, A is quasinormal provided A, is quasinormal for every prime idal p of A, We call a ring A locally quasinormal if A satisfies the last condition. We give, in the second section, some conditions for A to be locally quasinormal. A main result is Theorem 2.12. As a corollary we prove that A is locally quasinormal if and only if A is seminormal and locally or-closed (cf. 2.13). The first author expresses his gratitude to Professor T. Asanuma, who pointed out to him a significance of the u-closedness in studying quasinormal rings.