On strongly primary monoids and domains. Issue 9 (1st September 2020)

• Record Type: Journal Article
• Authors: Geroldinger, Alfred; Roitman, Moshe
• Is Part Of: Communications in algebra. Volume 48:Issue 9(2020)
• Page Start: 4085
• Page End: 4099
• Publication Date: 2020-09-01
• Subjects: Local tameness -- one-dimensional local domains -- primary monoids -- sets of distances -- sets of lengths; 13A05 -- 13F05 -- 20M13; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1755678
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence, one-dimensional local Mori domains are strongly primary. We prove among other results that if R is a domain such that the conductor ( R : R ̂ ) vanishes, then Λ ( R ) is finite; that is, there exists a positive integer k such that each nonzero nonunit of R is a product of at most k irreducible elements. Using this result, we obtain that every strongly primary domain is locally tame, and that a domain R is globally tame if and only if Λ ( R ) = ∞ . In particular, we answer Problem 38 of the open problem list by Cahen et al. in the affirmative. Many of our results are formulated for monoids.