Let A be an integral domain, E an A-module and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A\propto E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∝</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> be the trivial ring extension of A by E (denoted also <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A(+)E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> and called the idealization ring of E over A). We prove that if A satisfies ACC<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$_d$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>d</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> on ideals and E is divisible, then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A\propto E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∝</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> satisfies epi-ACC on ideals if and only if E satisfies epi-ACC on submodules. We prove that if E is torsion-free and divisible, then <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A\propto E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∝</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> satisfies epi-ACC on ideals (respectively is an isonoetherian ring) if and only if E satisfies epi-ACC on submodules (respectively is an isonoetherian module). We prove that if D is an integral domain with quotient field <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$K\ne D$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>≠</mo> <mi>D</mi> </mrow> </math></EquationSource> </InlineEquation>, then the ring <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$D\propto K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>∝</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> satisfies epi-ACC on ideals if and only if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$D\propto K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>∝</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> satisfies ACC<InlineEquation ID="IEq9"> <EquationSource Format="TEX">$_d$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>d</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> on ideals if and only if <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$D\propto K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>∝</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> is isonoetherian if and only if D is a semi-local PID.

Trivial ring extension with epi-ACC on ideals

期刊论文
发表日期：

2025年6月25日

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Mohamed Khalifa

摘要

Let A be an integral domain, E an A-module and $A\propto E$ $A \propto E$ be the trivial ring extension of A by E (denoted also $A(+)E$ $A (+) E$ and called the idealization ring of E over A). We prove that if A satisfies ACC $_d$ $_{d}$ on ideals and E is divisible, then $A\propto E$ $A \propto E$ satisfies epi-ACC on ideals if and only if E satisfies epi-ACC on submodules. We prove that if E is torsion-free and divisible, then $A\propto E$ $A \propto E$ satisfies epi-ACC on ideals (respectively is an isonoetherian ring) if and only if E satisfies epi-ACC on submodules (respectively is an isonoetherian module). We prove that if D is an integral domain with quotient field $K\ne D$ $K \neq D$ , then the ring $D\propto K$ $D \propto K$ satisfies epi-ACC on ideals if and only if $D\propto K$ $D \propto K$ satisfies ACC $_d$ $_{d}$ on ideals if and only if $D\propto K$ $D \propto K$ is isonoetherian if and only if D is a semi-local PID.