Factoring Nonnil Ideals into Prime and Invertible Ideals

For a commutative ring R, let Nil(R) be the set of all nilpotent elements of R, Z(R) be the set of all zero divisors of R, T (R) be the total quotient ring of R, and H = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. For a ring R ∈ H, let φ : T (R) −→ RNil(R) such that φ(a/b) = a/b for every a ∈ R and b ∈ R\Z(R). A ring R is called a ZPUI ring if every proper ideal of R can be written as a finite product of invertible and prime ideals of R. In this paper, we give a generalization of the concept of ZPUI domains which was extensively studied by Olberding to the context of rings that are in the class H. Let R ∈ H. If every nonnil ideal of R can be written as a finite product of invertible and prime ideals of R, then R is called a nonnil-ZPUI ring; if every nonnil ideal of φ(R) can be written as a finite product of invertible and prime ideals of φ(R), then R is called a nonnil-φ-ZPUI ring. We show that the theory of φ-ZPUI rings resembles that of ZPUI domains.