Structure and characterizations of maximal non-treed rings

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℋ</m:mi> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℋ</m:mi> </m:math> {H} . Given a ring extension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>⊆</m:mo> <m:mi>T</m:mi> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">ℋ</m:mi> </m:mrow> </m:math> {R} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>Nil</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>Z</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {Nil(R)=Z(R)} , then R is a maximal non-treed ring if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>R</m:mi> <m:mi>red</m:mi> </m:msub> </m:math> {Rred} 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.