A new semistar operation on a commutative ring and its applications. Issue 9 (1st September 2020)

• Record Type: Journal Article
• Authors: Zhou, De Chuan; Kim, Hwankoo; Wang, Fang-Gui; Chen, Dan
• Is Part Of: Communications in algebra. Volume 48:Issue 9(2020)
• Page Start: 3973
• Page End: 3988
• Publication Date: 2020-09-01
• Subjects: Cartan-Eilenberg-Bass theorem -- Hilbert basis theorem -- Krull intersection theorem -- Krull's principal ideal theorem -- Lucas module -- τq-Noetherian ring -- q-operation -- semiregular ideal -- the ring of finite fractions; 13C99 -- 13A15; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1753060
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: In this article, a new semistar operation, called the q -operation, on a commutative ring R is introduced in terms of the ring Q 0 ( R ) of finite fractions. It is defined as the map q : F q ( R ) → F q ( R ) by A ↦ A q : = { x ∈ Q 0 ( R ) | there exists some finitely generated semiregular ideal J of R such that J x ⊆ A } for any A ∈ F q ( R ), where F q ( R ) denotes the set of nonzero R -submodules of Q 0 ( R ) . The main superiority of this semistar operation is that it can also act on R -modules. We can also get a new hereditary torsion theory τq induced by a (Gabriel) topology { I | I is an ideal of R with I q = R q } . Based on the existing literature of τq -Noetherian rings by Golan and Bland et al., in terms of the q -operation, we can study them in more detailed and deep module-theoretic point of view, such as τq -analog of the Hilbert basis theorem, Krull's principal ideal theorem, Cartan-Eilenberg-Bass theorem, and Krull intersection theorem.