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