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- W3170272470 abstract "Abstract Let G be a group. Let R be a G -graded commutative ring and let M be a graded R -module. A proper graded submodule Q of M is called a graded quasi-primary submodule if whenever <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>r</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>h</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> {rin h(R)} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>h</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {min h(M)} with <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>r</m:mi> <m:mo></m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>Q</m:mi> </m:mrow> </m:math> {rmin Q} , then either <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>r</m:mi> <m:mo>∈</m:mo> <m:mi>Gr</m:mi> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>Q</m:mi> <m:msub> <m:mo>:</m:mo> <m:mi>R</m:mi> </m:msub> <m:mi>M</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {rinoperatorname{Gr}((Q:_{R}M))} or <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>Gr</m:mi> <m:mi>M</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>Q</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {minoperatorname{Gr}_{M}(Q)} . The graded quasi-primary spectrum <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mrow> <m:mi>qp</m:mi> <m:mo>.</m:mo> <m:mi>Spec</m:mi> </m:mrow> <m:mi>g</m:mi> </m:msub> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathop{rm qp.Spec}nolimits_{g}(M)} is defined to be the set of all graded quasi-primary submodules of M . In this paper, we introduce and study a topology on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mrow> <m:mi>qp</m:mi> <m:mo>.</m:mo> <m:mi>Spec</m:mi> </m:mrow> <m:mi>g</m:mi> </m:msub> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathop{rm qp.Spec}nolimits_{g}(M)} , called the quasi-Zariski topology, and investigate the properties of this topology and some conditions under which <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>qp</m:mi> <m:mo>.</m:mo> <m:mi>Spec</m:mi> </m:mrow> <m:mi>g</m:mi> </m:msub> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>.</m:mo> <m:msup> <m:mi>τ</m:mi> <m:mi>g</m:mi> </m:msup> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:math> {(mathop{rm qp.Spec}nolimits_{g}(M),q.tau^{g})} is a Noetherian, spectral space." @default.
- W3170272470 created "2021-06-22" @default.
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- W3170272470 date "2023-10-04" @default.
- W3170272470 modified "2023-10-18" @default.
- W3170272470 title "The quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring" @default.
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- W3170272470 doi "https://doi.org/10.1515/gmj-2023-2075" @default.
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