论文标题
在主要理想方面的分裂性和上升的链条条件较弱
Divisibility and a weak ascending chain condition on principal ideals
论文作者
论文摘要
如果每个非零的$ r $ raunit因子在不查看中，则整体域$ r $是原子质。此外，如果每个增加的主要理想顺序（在包含在内）从一个点开始，则整体域$ r $满足主要理想的上升链条件（ACCP）。尽管不难验证满足ACCP的每个积分域都是原子的，但是众所周知，难以构建不满足ACCP的原子域的例子。此类示例中的第一个是由A. grams于1974年构建的。在本文中，我们深入研究了不满足ACCP的原子域类。为了更好地理解这一类，我们介绍了弱ACCP域的概念，该域概括了满足ACCP的整体域的概念。 D. D. Anderson，D。F。Anderson和M. Zafrullah在1990年引入了强烈的原子领域。事实证明，每个弱ACCP领域都是强烈的原子原子，因此我们对我们的兴趣类别介绍了分类学分类：ACCP暗示弱-ACCP，暗示着强烈的原子性，这暗示着强烈的原子性，这意味着ATOMICTOMICTOMICITICITITYATOMICITICITY。我们研究了这种含义链，特别强调了弱ACCP特性。这使我们能够提供不满足ACCP的原子领域的新示例。
An integral domain $R$ is atomic if each nonzero nonunit of $R$ factors into irreducibles. In addition, an integral domain $R$ satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. Although it is not hard to verify that every integral domain satisfying ACCP is atomic, examples of atomic domains that do not satisfy ACCP are notoriously hard to construct. The first of such examples was constructed by A. Grams back in 1974. In this paper we delve into the class of atomic domains that do not satisfy ACCP. To better understand this class, we introduce the notion of weak-ACCP domains, which generalizes that of integral domains satisfying ACCP. Strongly atomic domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. It turns out that every weak-ACCP domain is strongly atomic, and so we introduce a taxonomic classification on our class of interest: ACCP implies weak-ACCP, which implies strong atomicity, which implies atomicity. We study this chain of implications, putting special emphasis on the weak-ACCP property. This allows us to provide new examples of atomic domains that do not satisfy ACCP.
