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- W3119977316 abstract "Abstract Let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mo stretchy=false>(</m:mo><m:mtext>Γ</m:mtext><m:mo>,</m:mo><m:mo>≤</m:mo><m:mo stretchy=false>)</m:mo></m:math> ({mathrm{Gamma}},le ) be a strictly ordered monoid, and let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:msup><m:mrow><m:mtext>Γ</m:mtext></m:mrow><m:mtext>⁎</m:mtext></m:msup><m:mo>=</m:mo><m:mtext>Γ</m:mtext><m:mo></m:mo><m:mo stretchy=false>{</m:mo><m:mn>0</m:mn><m:mo stretchy=false>}</m:mo></m:math> {{mathrm{Gamma}}}^{ast }left={mathrm{Gamma}}backslash {0} . Let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mi>D</m:mi><m:mo>⊆</m:mo><m:mi>E</m:mi></m:math> Dsubseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D . Set <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block><m:mtable displaystyle=true columnalign=left><m:mtr columnalign=left><m:mtd columnalign=left><m:mrow><m:mi>D</m:mi><m:mo>+</m:mo><m:mrow><m:mo stretchy=true>〚</m:mo><m:mrow><m:msup><m:mrow><m:mi>E</m:mi></m:mrow><m:mrow><m:msup><m:mrow><m:mtext>Γ</m:mtext></m:mrow><m:mo>⁎</m:mo></m:msup><m:mo>,</m:mo><m:mo>≤</m:mo></m:mrow></m:msup></m:mrow><m:mo stretchy=true>〛</m:mo></m:mrow><m:mo>≔</m:mo><m:mrow><m:mfenced open={ close=}><m:mrow><m:mi>f</m:mi><m:mo>∈</m:mo><m:mrow><m:mo stretchy=true>〚</m:mo><m:mrow><m:msup><m:mrow><m:mi>E</m:mi></m:mrow><m:mrow><m:mtext>Γ</m:mtext><m:mo>,</m:mo><m:mo>≤</m:mo></m:mrow></m:msup></m:mrow><m:mo stretchy=true>〛</m:mo></m:mrow><m:mspace width=0.15em /><m:mo>|</m:mo><m:mspace width=0.2em /><m:mi>f</m:mi><m:mo stretchy=false>(</m:mo><m:mn>0</m:mn><m:mo stretchy=false>)</m:mo><m:mo>∈</m:mo><m:mi>D</m:mi></m:mrow></m:mfenced></m:mrow><m:mspace width=.5em /><m:mtext>and</m:mtext></m:mrow></m:mtd></m:mtr><m:mtr columnalign=left><m:mtd columnalign=left><m:mspace width=0.2em /><m:mrow><m:mi>D</m:mi><m:mo>+</m:mo><m:mrow><m:mo stretchy=true>〚</m:mo><m:mrow><m:msup><m:mrow><m:mi>I</m:mi></m:mrow><m:mrow><m:msup><m:mrow><m:mo>Γ</m:mo></m:mrow><m:mo>⁎</m:mo></m:msup><m:mo>,</m:mo><m:mo>≤</m:mo></m:mrow></m:msup></m:mrow><m:mo stretchy=true>〛</m:mo></m:mrow><m:mo>≔</m:mo><m:mrow><m:mfenced open={ close=}><m:mrow><m:mi>f</m:mi><m:mo>∈</m:mo><m:mrow><m:mo stretchy=true>〚</m:mo><m:mrow><m:msup><m:mrow><m:mi>D</m:mi></m:mrow><m:mrow><m:mtext>Γ</m:mtext><m:mo>,</m:mo><m:mo>≤</m:mo></m:mrow></m:msup></m:mrow><m:mo stretchy=true>〛</m:mo></m:mrow><m:mspace width=0.15em /><m:mo>|</m:mo><m:mspace width=0.2em /><m:mi>f</m:mi><m:mo stretchy=false>(</m:mo><m:mi>α</m:mi><m:mo stretchy=false>)</m:mo><m:mo>∈</m:mo><m:mi>I</m:mi><m:mo>,</m:mo><m:mspace width=.5em /><m:mtext>for</m:mtext><m:mspace width=.25em /><m:mtext>all</m:mtext><m:mspace width=.5em /><m:mi>α</m:mi><m:mo>∈</m:mo><m:msup><m:mrow><m:mtext>Γ</m:mtext></m:mrow><m:mtext>⁎</m:mtext></m:msup></m:mrow></m:mfenced></m:mrow><m:mo>.</m:mo></m:mrow></m:mtd></m:mtr></m:mtable></m:math> begin{array}{l}D+[kern-2pt[ {E}^{{{mathrm{Gamma}}}^{ast },le }]kern-2pt] := left{fin [kern-2pt[ {E}^{{mathrm{Gamma}},le }]kern-2pt] hspace{0.15em}|hspace{0.2em}f(0)in Dright}hspace{.5em}text{and} hspace{0.2em}D+[kern-2pt[ {I}^{{Gamma }^{ast },le }]kern-2pt] := left{fin [kern-2pt[ {D}^{{mathrm{Gamma}},le }]kern-2pt] hspace{0.15em}|hspace{0.2em}f(alpha )in I,hspace{.5em}text{for}hspace{.25em}text{all}hspace{.5em}alpha in {{mathrm{Gamma}}}^{ast }right}.end{array} In this paper, we give necessary conditions for the rings <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mi>D</m:mi><m:mo>+</m:mo><m:mrow><m:mo stretchy=true>〚</m:mo><m:mrow><m:msup><m:mrow><m:mi>E</m:mi></m:mrow><m:mrow><m:msup><m:mrow><m:mtext>Γ</m:mtext></m:mrow><m:mo>⁎</m:mo></m:msup><m:mo>,</m:mo><m:mo>≤</m:mo></m:mrow></m:msup></m:mrow><m:mo stretchy=true>〛</m:mo></m:mrow></m:math> D+[kern-2pt[ {E}^{{{mathrm{Gamma}}}^{ast },le }]kern-2pt] to be Noetherian when <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mo stretchy=false>(</m:mo><m:mtext>Γ</m:mtext><m:mo>,</m:mo><m:mo>≤</m:mo><m:mo stretchy=false>)</m:mo></m:math> ({mathrm{Gamma}},le ) is positively ordered, and sufficient conditions for the rings <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mi>D</m:mi><m:mo>+</m:mo><m:mrow><m:mo stretchy=true>〚</m:mo><m:mrow><m:msup><m:mrow><m:mi>E</m:mi></m:mrow><m:mrow><m:msup><m:mrow><m:mtext>Γ</m:mtext></m:mrow><m:mo>⁎</m:mo></m:msup><m:mo>,</m:mo><m:mo>≤</m:mo></m:mrow></m:msup></m:mrow><m:mo stretchy=true>〛</m:mo></m:mrow></m:math> D+[kern-2pt[ {E}^{{{mathrm{Gamma}}}^{ast },le }]kern-2pt] to be Noetherian when <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mo stretchy=false>(</m:mo><m:mtext>Γ</m:mtext><m:mo>,</m:mo><m:mo>≤</m:mo><m:mo stretchy=false>)</m:mo></m:math> ({mathrm{Gamma}},le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mi>D</m:mi><m:mo>+</m:mo><m:mrow><m:mo stretchy=true>〚</m:mo><m:mrow><m:msup><m:mrow><m:mi>I</m:mi></m:mrow><m:mrow><m:msup><m:mrow><m:mo>Γ</m:mo></m:mrow><m:mo>⁎</m:mo></m:msup><m:mo>,</m:mo><m:mo>≤</m:mo></m:mrow></m:msup></m:mrow><m:mo stretchy=true>〛</m:mo></m:mrow></m:math> D+[kern-2pt[ {I}^{{Gamma }^{ast },le }]kern-2pt] to be Noetherian when <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mo stretchy=false>(</m:mo><m:mtext>Γ</m:mtext><m:mo>,</m:mo><m:mo>≤</m:mo><m:mo stretchy=false>)</m:mo></m:math> ({mathrm{Gamma}},le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mi>D</m:mi><m:mo>+</m:mo><m:mo stretchy=false>(</m:mo><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:mi>…</m:mi><m:mo>,</m:mo><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:msub><m:mo stretchy=false>)</m:mo><m:mi>E</m:mi><m:mo stretchy=false>[</m:mo><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:mi>…</m:mi><m:mo>,</m:mo><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:msub><m:mo stretchy=false>]</m:mo></m:math> D+({X}_{1},ldots ,{X}_{n})E{[}{X}_{1},ldots ,{X}_{n}] and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mi>D</m:mi><m:mo>+</m:mo><m:mo stretchy=false>(</m:mo><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:mi>…</m:mi><m:mo>,</m:mo><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:msub><m:mo stretchy=false>)</m:mo><m:mi>I</m:mi><m:mo stretchy=false>[</m:mo><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:mi>…</m:mi><m:mo>,</m:mo><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:msub><m:mo stretchy=false>]</m:mo></m:math> D+({X}_{1},ldots ,{X}_{n})I{[}{X}_{1},ldots ,{X}_{n}] to be Noetherian." @default.
- W3119977316 created "2021-01-18" @default.
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- W3119977316 date "2020-01-01" @default.
- W3119977316 modified "2023-09-23" @default.
- W3119977316 title "Noetherian properties in composite generalized power series rings" @default.
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- W3119977316 doi "https://doi.org/10.1515/math-2020-0103" @default.
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