Maximal Non $$\phi $$-Chained Rings and Maximal Non Chained Rings

Abstract

Let R be a commutative ring with unity. The notion of maximal non chained subrings of a ring and maximal non $$\phi $$ -chained subrings of a ring is introduced which generalizes the concept of maximal non valuation subrings of a domain. A ring R is said to be a maximal non chained (resp., $$\phi $$ -chained) subring of S if R is a proper subring of S, R is not a chained (resp., $$\phi $$ -chained) ring and every subring of S which contains R properly is a chained (resp., $$\phi $$ -chained) ring. We study the properties and characterizations of a maximal non chained ( $$\phi $$ -chained) subring of a ring. Examples of a maximal non $$\phi $$ -chained subring which is not a maximal non chained subring and a maximal non chained subring which is not a maximal non $$\phi $$ -chained subring are also given to strengthen the concept.