Maximal non-Prüfer and maximal non--Prüfer rings

Abstract

Let R be a commutative ring with unity. Let $\mathcal{H}$ denotes the set of all rings R such that $\text{Nil}(\mathrm{R})$ is a divided prime ideal. The notion of maximal non-Prüfer ring and maximal non-$\varphi$-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-$\varphi$-Prüfer subring of S if R is not a $\varphi$-Prüfer ring but every subring of S which contains R properly is a $\varphi$-Prüfer ring. We study the properties of maximal non-Prüfer subrings and maximal non-$\varphi$-Prüfer subrings of a ring in class $\mathcal{H}.$ Characterizations of a ring in class $\mathcal{H}$ to be a maximal non-Prüfer ring and maximal non-$\varphi$-Prüfer ring are given. Examples of a maximal non-$\varphi$-Prüfer subring which is not a maximal non-Prüfer subring and a maximal non-Prüfer subring which is not a maximal non-$\varphi$-Prüfer subring are also given to strengthen the concept.

Keywords:

• Maximal non-Prüfer ring
• maximal non-$\varphi$-Prüfer ring
• Prüfer ring
• $\varphi$-Prüfer ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13B02
• 13B22
• 13B25
• 13B30
• 13A18

Additional information

Funding

Atul Gaur was supported by the MATRICS grant from DST-SERB India, No. MTR/2018/000707. Rahul Kumar was supported by the SRF grant from UGC India, Sr. No. 2061440976.