Structure and characterizations of maximal non-treed rings

Abstract Let ℋ H denote the class of all commutative rings with identity whose nilradical is a divided prime ideal. In this paper, we introduce and study the concept of maximal non-treed subrings, extending the concept from integral domains to the broader class ℋ H. Given a ring extension R ⊆ T R T, we say that R is a maximal non-treed subring of T if R is not a treed ring, but every proper subring of T properly containing R is a treed ring. We provide several characterizations of such rings and investigate their structural properties. In particular, we show that if R ∈ ℋ R and Nil ⁢ (R) = Z ⁢ (R) Nil (R) =Z (R), then R is a maximal non-treed ring if and only if R red Rₑ₄₃ is a maximal non-treed subring of its total quotient ring. We further explore connections with φ-QQR rings and ring extensions via amalgamated algebras. Several examples are provided to illustrate and support the developed theory.