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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
multiplicative filter
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(Example)
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documentclass[12pt]article pagestyleempty setlengthpaperwidth8.5in setlengthpaperheight11in par setlengthtopmargin0.00in setlengthheadsep0.00in setlengthheadheight0.00in setlengthevensidemargin0.00in setlengthoddsidemargin0.00in setlengthtextwidth6.5in setlengthtextheight9.00in setlengthvoffset0.00in setlengthhoffset0.00in setlengthmarginparwidth0.00in setlengthmarginparsep0.00in setlengthparindent0.00in setlengthparskip0.15in par usepackagehtml par usepackageamssymb usepackageamsmath usepackageamsfonts par newedcommandsep: par begindocument par For any htmladdnormallinkringhttp://planetmath.org/encyclopedia/UnitalRing.html $A$ , any set $S\subset A$ and any element $x\in A$ , we use the notation $$(S:x):=\{ a\in A\ ax\in S\}$$ par Let $A$ be a htmladdnormallinkcommutative ringhttp://planetmath.org/encyclopedia/CommutativeRing.html with
htmladdnormallinkunityhttp://planetmath.org/encyclopedia/CharacterizationOfUnity.html, and let $\mathcal{I}(A)$ be the set of all htmladdnormallinkidealshttp://planetmath.org/encyclopedia/PrincipalIdeal4.html of $A$ . beginitemize item A emphMultiplicative Filter of $A$ is a htmladdnormallinkfilterhttp://planetmath.org/encyclopedia/PrincipalElement.html $\mathcal{F}$ on $\mathcal{I}(A)$ such that $I,J\in\mathcal{F}\Rightarrow IJ\in\mathcal{F}$ . item A emphGabriel Filter of $A$ is a filter $\mathcal{F}$ on $\mathcal{I}(A)$ such that par $$ [I\in\mathcal{F},J\in\mathcal{I}(A)\textrm{ and }\forall x\in I,(J:x)\in\mathcal{F}]\Rightarrow J\in\mathcal{F} $$ enditemize par Note that Gabriel Filters are also
htmladdnormallinkMultiplicativehttp://planetmath.org/encyclopedia/CompletelyMultiplicativeFunction.html Filters. par enddocument
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"multiplicative filter" is owned by jocaps.
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(view preamble | get metadata)
| Also defines: |
Gabriel Filter, Multiplicative Filter |
This object's parent.
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Cross-references: multiplicative, filter, ideals, unity, commutative ring, ring
This is version 3 of multiplicative filter, born on 2007-03-06, modified 2007-03-06.
Object id is 9040, canonical name is MultiplicativeFilters.
Accessed 1868 times total.
Classification:
| AMS MSC: | 54A99 (General topology :: Generalities :: Miscellaneous) | | | 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous) |
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Pending Errata and Addenda
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