A characterization of Krull domains in terms of their factor rings. Issue 3 (4th March 2023)

• Record Type: Journal Article
• Authors: Chang, Gyu Whan; Kim, Hwankoo
• Is Part Of: Communications in algebra. Volume 51:Issue 3(2023)
• Page Start: 1280
• Page End: 1292
• Publication Date: 2023-03-04
• Subjects: Domain of Krull type -- generalized Krull domain -- integral domain of finite t-character -- Krull domain -- strong Mori domain -- weakly Krull domain -- w-operation; 13A15 -- 13F05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2022.2134404
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let D be an integral domain, w be the so-called w -operation on D, and GV ( D ) be the set of nonzero finitely generated ideals J of D with J − 1 = D . Let I be a w -ideal of D, D / I be the factor ring of D modulo I, E ( D / I ) be the injective envelope of D / I, and ( D / I ) w = { x ∈ E ( D / I ) | J x ⊆ D / I for some J ∈ GV ( D ) } . Recently Zhou, Kim, and Hu showed that ( D / I ) w is a commutative ring with identity containing D / I . The Krull domains are a w -analog of the Dedekind domains in the sense that D is a Krull domain if and only if every nonzero ideal of D is w -invertible. It is well known that D is a Dedekind domain if and only if D / aD is a zero-dimensional semilocal principal ideal ring (PIR) for all nonzero nonunits a ∈ D, if and only if D / I is a zero-dimensional semilocal PIR for all nonzero ideals I of D . In this paper, we prove that D is a Krull domain if and only if ( D / a D ) w is a zero-dimensional semilocal PIR for any nonzero nonunit a of D, if and only if ( D / I ) w is a zero-dimensional semilocal PIR for any w -ideal I of D . We also study when ( D / I ) w is Noetherian, Artinian, or a (zero-dimensional) semilocal Bézout ring for all nonzero principal ideals or w -ideals I of D .