Maximal non $\lambda $-subrings

summary:Let $R$ be a commutative ring with unity. The notion of maximal non $\lambda $-subrings is introduced and studied. A ring $R$ is called a maximal non $\lambda $-subring of a ring $T$ if $R\subset T$ is not a $\lambda $-extension, and for any ring $S$ such that $R\subset S\subseteq T$, $S\subseteq T$ is a $\lambda $-extension. We show that a maximal non $\lambda $-subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D + M$ construction is a maximal non $\lambda $-domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non $\lambda $-subring. If $R$ is a maximal non $\lambda $-subring of a field $K$, where $R$ is integrally closed in $K$, then $K$ is the quotient field of $R$ and $R$ is a Prüfer domain. The equivalence of a maximal non $\lambda $-domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non $\lambda $-subrings of a field