Research Projects

Badawi introduced the concept of ϕ-rings in 1999, which is a generalization of domains to rings. This concept has been a fascination for mathematicians for a long time and has been successfully applied to solve seemingly unrelated and challenging problems in Mathematics. In the 21st century, algebraists began generalizing the concept of well-known domains through the class H, which consists of all commutative rings whose nilradical is a divided prime ideal. S. Visweswaran's observation in 1990 led to the introduction of the concept of maximal non-Noetherian subring of a domain. He proved that if R is non-Noetherian and all other subrings of S that properly contain R are Noetherian, then R is called a maximal non-Noetherian subring of S. This work has led to many studies on the extension of domains with intermediate domains satisfying various ring-theoretic properties P. The theory of maximal non-P-subrings of an integral domain is well established in commutative ring theory and has been used by researchers to study multiplicative ideal theory. This proposal aims to link ϕ-rings and maximal non-P-subrings of an integral domain, helping researchers find new ring extensions where intermediate rings are not domains but satisfy a fixed ring theoretic property. The research will focus on maximal non-ϕ-subrings of a ring in class H, filling the gap between these two concepts and helping other area researchers study generalizations of integral domains via ϕ-rings. This project has many applications in topology and algebraic geometry, and it will also help linear algebra people study matrices over ϕ-rings.