整数值多项式环的整除性
Divisibility in rings of integer-valued polynomials
摘要：
在这篇文章中，我们讨论了由整数值多项式组成的环中不可约整除性的各个方面。一个整环称为原子整环，如果每个非零非单位因子都是不可约的。不满足主理想上的升链条件（ACCP）的原子域是一种难以捉摸的域，自从A. 1974年的克。在这里，我们展示了第一类原子环的整数值多项式没有ACCP。一个整环被称为有限分解域（FFD），如果它同时是原子和idf域（即，每一个非零元素都只能被至多到联合的200个不可约元素整除）。我们证明了一个环是一个FFD当且仅当它的整数值多项式环是一个FFD。此外，我们还证明，一般来说，从整环到其整值多项式环的转移既不是原子的，也不是idf域的。在同一类整数值多项式环中，我们考虑了根据不可约整除性定义的进一步性质，包括Cohen-Kaplansky和Furstenberg。
英文摘要：
In this paper, we address various aspects of divisibility by irreducibles in rings consisting of integer-valued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not satisfy the ascending chain condition on principal ideals (ACCP) have proved to be elusive, and not many of them have been found since the first one was constructed by A. Grams in 1974. Here we exhibit the first class of atomic rings of integer-valued polynomials without the ACCP. An integral domain is called a finite factorization domain (FFD) if it is simultaneously atomic and an idf-domain (i.e., every nonzero element is divisible by only finitely many irreducibles up to associates). We prove that a ring is an FFD if and only if its ring of integer-valued polynomials is an FFD. In addition, we show that neither being atomic nor being an idf-domain transfer, in general, from an integral domain to its ring of integer-valued polynomials. In the same class of rings of integer-valued polynomials, we consider further properties that are defined in terms of divisibility by irreducibles, including being Cohen-Kaplansky and being Furstenberg.
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