ON φ-ALMOST BEZOUT RINGS AND φ-ALMOST PRUFER RINGS

The purpose of this paper is to introduce some new classes of rings that are closely related to the classes of almost Bezout domains and almost Prufer domains. Suppose that H = {R vertical bar R is a commutative ring with 1 not equal 0 and Nil(R) is a divided prime ideal of R}. Let R is an element of T(R) be the total qoutient ring of R, and set phi : T(R) -> R-Nil(R) such that phi(a/b) = a/b for every a is an element of R and b is an element of R \ Z(R). Then phi is a ring homomorphism from T(R) into R-Nil(R) and phi resticted to R is also a ring homomorphism from R into R-Nil(R) given by phi(x) = x/1 for every x is an element of R. An ideal I of R is phi-invertible if phi(I) is invertible ideal of phi(R). A ring R is a phi-almost Bezout ring (phi-AB ring) (respectively, phi-almost Prufer ring (phi-AP ring)) if for nonnil elements a, b of R, There exists an n = n(a, b) with (a(n), b(n)) is a nonnil principal (respectively, phi-invertible) ideal of R. This paper is devoted to study the phi-AB rings and phi-AP rings.