整环与IDF性质
Integral domains and the IDF property
摘要：
一个整环$D$称为不可约因子有限域（idf-domain），如果$D$的每个非零元素都有1/2个不可约因子直到其相伴. IDF域的研究可以追溯到七十年代。在本文中，我们调查的各个方面的IDF属性。2009年，P.~ Malcolmson和F. Okoh证明了从整环到相应的多项式环，IDF性质并不上升，回答了D. D.安德森，D. F.安德森和二十年前的第二位作者。在这里，我们证明了在PSP-域类中的IDF属性上升，推广了已知的结果（也是由Malcolmson和Okoh），IDF属性上升的GCD-域类。我们特别强调了IDF-domains，其中每个非单位都可以被一个不可约整除，我们称之为TIDF-domains，我们还考虑了PIDF-domains，它们形成了Malcolmson和Okoh在2006年引入的一类特殊的IDF-domains。我们研究了在取多项式环和局部化下TIDF和PIDF的性质。我们还深入研究了它们在么半群整环和$D+M$结构下的行为
英文摘要：
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
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