On Krull rings with zero divisors. Issue 1 (2nd January 2021)

• Record Type: Journal Article
• Authors: Chang, Gyu Whan; Kang, Byung Gyun
• Is Part Of: Communications in algebra. Volume 49:Issue 1(2021)
• Page Start: 207
• Page End: 215
• Publication Date: 2021-01-02
• Subjects: Krull ring -- rank one DVR -- t-operation; 13A15 -- 13F05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1797071
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Let R be a commutative ring with identity and X r 1 ( R ) the set of regular height one prime ideals of R . We will show that R is a Krull ring if and only if each regular prime ideal of R contains a t -invertible prime ideal, if and only if (1) R = ∩ P ∈ X r 1 ( R ) R [ P ] and it has finite character and (2) ( R [ P ], [ P ] R [ P ] ) is a rank one DVR for each P ∈ X r 1 ( R ) . It is also shown that { ( R [ P ], [ P ] R [ P ] ) | P ∈ X r 1 ( R ) } is the unique defining family for a Krull ring R .