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Let  R  be a commutative ring with unity. Letdenotes the set of all rings  R  such thatis 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 classCharacterizations of a ring in classto 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.

Maximal non-Prüfer and maximal non- -Prüfer rings