On ϕ-piecewise Noetherian rings. Issue 3 (21st October 2020)

• Record Type: Journal Article
• Authors: Khalfi, Abdelhaq El; Kim, Hwankoo; Mahdou, Najib
• Is Part Of: Communications in algebra. Volume 49:Issue 3(2021)
• Page Start: 1324
• Page End: 1337
• Publication Date: 2020-10-21
• Subjects: ϕ-piecewise Noetherian rings -- ϕ-piecewise w-Noetherian rings -- trivial ring extension; Primary 13B99 -- Secondary 13A15 -- 13G05 -- 13B21; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1834571
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w -Noetherian domains. Let H = { R | R is a commutative ring and Nil ( R ) is a divided prime ideal of R } . Let R ∈ H be a ring with total quotient ring T ( R ), and set ϕ : T ( R ) → R Nil ( R ) such that ϕ ( a b ) = a b for every a ∈ R and every b ∈ R ∖ Z ( R ) . Then ϕ is a ring homomorphism from T ( R ) into R Nil ( R ), and ϕ restricted to R is also a ring homomorphism from R into R Nil ( R ) given by ϕ ( x ) = x 1 for every x ∈ R . At this setting, we introduce the notions of ϕ -piecewise Noetherian rings and ϕ -piecewise w -Noetherian rings and we show that the theories of ϕ -piecewise Noetherian (resp. ϕ -piecewise w -Noetherian) rings resemble those of piecewise Noetherian (resp. piecewise w -Noetherian) domains.