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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation> be an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varvec{m}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">m</mi> </mrow> </math></EquationSource> </InlineEquation>-system of a ring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varvec{P}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">P</mi> </mrow> </math></EquationSource> </InlineEquation> a submodule of a right <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\varvec{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> </math></EquationSource> </InlineEquation>-module <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\varvec{M}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation>. This paper presents the notion of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-prime submodule and provides some properties and equivalent definitions. We define <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-multiplication right module and prove that in multiplication (<InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-multiplication) right <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\varvec{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> </math></EquationSource> </InlineEquation>-module <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\varvec{M}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation>, the ideal <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(\varvec{(P:}_{\varvec{R}}\varvec{M)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">P</mi> <mo mathvariant="bold">:</mo> </mrow> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> </msub> <mrow> <mi mathvariant="bold-italic">M</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a right <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-prime ideal of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(\varvec{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(\varvec{P}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">P</mi> </mrow> </math></EquationSource> </InlineEquation> is an <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-prime submodule of <InlineEquation ID="IEq17"> <EquationSource Format="TEX">\(\varvec{M}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we give an <InlineEquation ID="IEq18"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-version of the prime avoidance lemma. Furthermore, we define <InlineEquation ID="IEq19"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-finite and <InlineEquation ID="IEq20"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-Noetherian right modules following the definitions in Abouhalaka, A (Mediterr. J. Math. 21(2), 43 <CitationRef CitationID="CR1">2024</CitationRef>). We prove that a multiplication finitely generated right <InlineEquation ID="IEq21"> <EquationSource Format="TEX">\(\varvec{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> </math></EquationSource> </InlineEquation>-module <InlineEquation ID="IEq22"> <EquationSource Format="TEX">\(\varvec{M}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq23"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-Noetherian if <InlineEquation ID="IEq24"> <EquationSource Format="TEX">\(\varvec{(N:}_{\varvec{R}}\varvec{M)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">N</mi> <mo mathvariant="bold">:</mo> </mrow> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> </msub> <mrow> <mi mathvariant="bold-italic">M</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an <InlineEquation ID="IEq25"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-prime ideal of <InlineEquation ID="IEq26"> <EquationSource Format="TEX">\(\varvec{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">R</mi> </mrow> </math></EquationSource> </InlineEquation>, for all submodules <InlineEquation ID="IEq27"> <EquationSource Format="TEX">\(\varvec{N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">N</mi> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq28"> <EquationSource Format="TEX">\(\varvec{M}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation>. In addition, we give some examples of right <InlineEquation ID="IEq29"> <EquationSource Format="TEX">\(\varvec{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">S</mi> </mrow> </math></EquationSource> </InlineEquation>-Noetherian rings.</p>

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S-PRIME RIGHT SUBMODULES AND AN S-VERSION OF PRIME AVOIDANCE

  • 期刊论文

  • 发表日期：

    2025年12月2日

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  • Alaa Abouhalaka

摘要

Let \(\varvec{S}\) S be an \(\varvec{m}\) m -system of a ring \(\varvec{R}\) R and \(\varvec{P}\) P a submodule of a right \(\varvec{R}\) R -module \(\varvec{M}\) M . This paper presents the notion of \(\varvec{S}\) S -prime submodule and provides some properties and equivalent definitions. We define \(\varvec{S}\) S -multiplication right module and prove that in multiplication ( \(\varvec{S}\) S -multiplication) right \(\varvec{R}\) R -module \(\varvec{M}\) M , the ideal \(\varvec{(P:}_{\varvec{R}}\varvec{M)}\) ( P : R M ) is a right \(\varvec{S}\) S -prime ideal of \(\varvec{R}\) R if and only if \(\varvec{P}\) P is an \(\varvec{S}\) S -prime submodule of \(\varvec{M}\) M . Moreover, we give an \(\varvec{S}\) S -version of the prime avoidance lemma. Furthermore, we define \(\varvec{S}\) S -finite and \(\varvec{S}\) S -Noetherian right modules following the definitions in Abouhalaka, A (Mediterr. J. Math. 21(2), 43 2024). We prove that a multiplication finitely generated right \(\varvec{R}\) R -module \(\varvec{M}\) M is \(\varvec{S}\) S -Noetherian if \(\varvec{(N:}_{\varvec{R}}\varvec{M)}\) ( N : R M ) is an \(\varvec{S}\) S -prime ideal of \(\varvec{R}\) R , for all submodules \(\varvec{N}\) N of \(\varvec{M}\) M . In addition, we give some examples of right \(\varvec{S}\) S -Noetherian rings.