Maximal non-Prüfer and maximal non--Prüfer rings

Abstract

Let R be a commutative ring with unity. Let H denotes the set of all rings R such that Nil(R) is a divided prime ideal. The notion of maximal non-Prüfer ring and maximal non-ϕ-Prüfer ring is introduced which generalize the concept of maximal non-Prüfer subrings of a field. A proper subring R of a ring S is said to be a maximal non-Prüfer subring of S if R is not a Prüfer ring but every subring of S which contains R properly is a Prüfer ring. A proper subring R of a ring S is said to be maximal non-ϕ-Prüfer subring of S if R is not a ϕ-Prüfer ring but every subring of S which contains R properly is a ϕ-Prüfer ring. We study the properties of maximal non-Prüfer subrings and maximal non-ϕ-Prüfer subrings of a ring in class H. Characterizations of a ring in class H to be a maximal non-Prüfer ring and maximal non-ϕ-Prüfer ring are given. Examples of a maximal non-ϕ-Prüfer subring which is not a maximal non-Prüfer subring and a maximal non-Prüfer subring which is not a maximal non-ϕ-Prüfer subring are also given to strengthen the concept.