On φ-Prüfer rings and φ-Bezout rings

Abstract. The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prüfer domains and Bezout domains. Let H = {R | R is a commutative ring with 1 6 = 0 and Nil(R) is a divided prime ideal of R}. Let R ∈ H, T (R) be the total quotient ring of R, and set φ: T (R) − → RNil(R) such that φ(a/b) = a/b for every a ∈ R and b ∈ R \ Z(R). Then φ is a ring homomorphism from T (R) into RNil(R), and φ restricted to R is also a ring homomorphism from R into RNil(R) given by φ(x) = x/1 for every x ∈ R. A nonnil ideal I of R is said to be φ-invertible if φ(I) is an invertible ideal of φ(R). If every finitely generated nonnil ideal of R is φ-invertible, then we say that R is a φ-Prüfer ring. Also, we say that R is a φ-Bezout ring if φ(I) is a principal ideal of φ(R) for every finitely generated nonnil ideal I of R. We show that the theories of φ-Prüfer and φ-Bezout rings resemble that of Prüfer and Bezout domains. 1