Divisibility in rings of integer-valued polynomials

被引:0
作者
Gotti, Felix [1 ]
Li, Bangzheng [2 ]
机构
[1] MIT, Dept Math, Cambridge, MA 02139 USA
[2] Christian Heritage Sch, Trumbull, CT 06611 USA
来源
NEW YORK JOURNAL OF MATHEMATICS | 2022年 / 28卷
关键词
integer-valued polynomials; atomic domain; ACCP; ascending chain condition on principal ideals; FFD; finite factorization domain; idf-domain; Furstenberg domain; atomicity; factorization theory; ALGEBRAIC-SETS; ATOMIC RINGS; DOMAINS; FACTORIZATION; ELASTICITY; EXTENSIONS;
D O I
暂无
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
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 being an idf-domain does not 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.
引用
收藏
页码:117 / 139
页数:23
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