论文标题
积分域和IDF属性
Integral domains and the IDF property
论文作者
论文摘要
如果$ d $的每个非零元素有限的许多不可减至的除法，则一个整体域$ d $称为不可减至的二型域域（IDF-DOMAIN）。 IDF域的研究可以追溯到七十年代。在本文中，我们研究了IDF属性的各个方面。 2009年，P.〜Malcolmson和F. Okoh证明，IDF财产并未从整体域上升到其相应的多项式环，回答了D. D. Anderson，D。F. Anderson和第二个作者二十年前提出的问题。在这里，我们证明了IDF属性在PSP域类中上升，从而概括了已知结果（也由Malcolmson和Okoh），IDF属性在GCD-Domains类中上升。我们特别强调了IDF-DOMAIN，其中每个非单元都可以被不可修复（我们称为TIDF-Domain）排除，我们还考虑PIDF-Domain，构成了Malcolmson和Okoh在2006年引入的特殊类IDF域。我们在TIDF和PIDF Properties中都在接受Polynomial and polynomial and polynomials和Polynomials Mings and and polynomials和Polynomials Mings and and polynomials and and polynomials。我们还深入研究了他们在Monoid域和$ D+M $构造下的行为
An integral domain $D$ is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of $D$ has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper, we investigate various aspects of the IDF property. In 2009, P.~Malcolmson and F. Okoh proved that the IDF property does not ascend from integral domains to their corresponding polynomial rings, answering a question posed by D. D. Anderson, D. F. Anderson, and the second author two decades before. Here we prove that the IDF property ascends in the class of PSP-domains, generalizing the known result (also by Malcolmson and Okoh) that the IDF property ascends in the class of GCD-domains. We put special emphasis on IDF-domains where every nonunit is divisible by an irreducible, which we call TIDF-domains, and we also consider PIDF-domains, which form a special class of IDF-domains introduced by Malcolmson and Okoh in 2006. We investigate both the TIDF and the PIDF properties under taking polynomial rings and localizations. We also delve into their behavior under monoid domain and $D+M$ constructions
