MCD-finite Domains and Ascent of IDF Property in Polynomial Extensions

ABSTRACT

An integral domain is said to have the IDF property when every non-zero element of it has only a finite number of non-associate irreducible divisors. A counterexample has already been found showing that the IDF property does not necessarily ascend in polynomial extensions. In this paper, we introduce a new class of integral domains, called MCD-finite domains, and show that for any domain D, D[X] is an IDF domain if and only if D is both IDF and MCD-finite. This result entails all the previously known sufficient conditions for the ascent of the IDF property. Our new characterization of polynomial domains with the IDF property enables us to use a different construction and build another counterexample which strengthen the previously known result on this matter.

KEYWORDS:

• Factorization
• IDF
• irreducible
• MCD
• polynomial extensions

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13B25
• 13F15
• 13F20
• 13G05

Acknowledgment

The authors would like to thank the referee whose careful reading and valuable comments improved the paper.