We extend the concept of maximal non-Noetherian subrings of a domain and define maximal non-nonnil-Noetherian subrings of a ring in class <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> is the set of all commutative rings with unity whose nilradical is a divided prime ideal. If <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R\subset T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>⊂</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> is a ring extension in class <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\text {Nil}(T)=\text {Nil}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Nil</mtext> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext>Nil</mtext> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, then R is a maximal non-nonnil-Noetherian subring of T if and only if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$R/\text {Nil}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo stretchy="false">/</mo> <mtext>Nil</mtext> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a maximal non-Noetherian subring of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T/\text {Nil}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">/</mo> <mtext>Nil</mtext> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Maximal Non-Nonnil-Noetherian Subrings of a Ring

期刊论文
发表日期：

2026年5月26日

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Anant Singh,
Atul Gaur,
Rahul Kumar

摘要

We extend the concept of maximal non-Noetherian subrings of a domain and define maximal non-nonnil-Noetherian subrings of a ring in class $\mathcal {H}$ $H$ , where $\mathcal {H}$ $H$ is the set of all commutative rings with unity whose nilradical is a divided prime ideal. If $R\subset T$ $R \subset T$ is a ring extension in class $\mathcal {H}$ $H$ such that $\text {Nil}(T)=\text {Nil}(R)$ $Nil (T) = Nil (R)$ , then R is a maximal non-nonnil-Noetherian subring of T if and only if $R/\text {Nil}(R)$ $R / Nil (R)$ is a maximal non-Noetherian subring of $T/\text {Nil}(R)$ $T / Nil (R)$ .