Generalizations of Prüfer rings and Bézout rings

• Published in: São Paulo Journal of Mathematical Sciences 2024, Vol.18 (1), p.126-141
• Main Authors: Kim, Hwankoo, Mahdou, Najib, Oubouhou, El Houssaine
• Format: Article
• Language: English
• Subjects: Homomorphisms; Mathematics; Mathematics and Statistics; Original Article; Rings (mathematics)
• ISSN: 1982-6907; 2316-9028; 2306-9028
• DOI: 10.1007/s40863-024-00413-y

The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prüfer domains, ϕ -Prüfer rings, Bézout domains, and ϕ -Bézout rings. Let G H = { R ∣ R is a commutative ring admitting a divided prime ideal P ⊆ Z ( R ) } . Let R ∈ G H and T ( R ) be the total ring of quotients of R . Define ϕ P : T ( R ) ⟶ R P by ϕ ( a / b ) = a / b for every a ∈ R and b ∈ R \ Z ( R ) . Then ϕ P is a ring homomorphism from T ( R ) into R P , and ϕ P restricted to R is also a ring homomorphism from R into R P given by ϕ P ( x ) = x / 1 for every x ∈ R . If Z ( ϕ P ( R ) ) = ϕ P ( P ) , then R is called a strongly ϕ P -ring. A P -ideal I of R (i.e., P ⊂ I ) is said to be ϕ P -invertible if ϕ P ( I ) is an invertible ideal of ϕ P ( R ) . If every finitely generated P -ideal of R is ϕ P -invertible, then we say that R is a ϕ P -Prüfer ring. We also say that R is a ϕ P -Bézout ring if ϕ P ( I ) is a principal ideal of ϕ P ( R ) for every finitely generated P -ideal I of R . We show that the theories of ϕ P -Prüfer and ϕ P -Bézout rings are similar to those of Prüfer and Bézout domains.