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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 , any set  and any element  , we use the notation
par Let  be a htmladdnormallinkcommutative ringhttp://planetmath.org/encyclopedia/CommutativeRing.html with htmladdnormallinkunityhttp://planetmath.org/encyclopedia/CharacterizationOfUnity.html, and let
 be the set of all htmladdnormallinkidealshttp://planetmath.org/encyclopedia/MaximalIdeal2.html of  . beginitemize item A emphMultiplicative Filter of  is a htmladdnormallinkfilterhttp://planetmath.org/encyclopedia/PrincipalElement.html
 on
 such that
 . item A emphGabriel Filter of  is a filter
 on
 such that par
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)
| 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 847 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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![$\displaystyle [Iinmathcal{F},Jinmathcal{I}(A)textrm{ and }forall xin I,(J:x)inmathcal{F}]Rightarrow Jinmathcal{F} $](http://images.planetmath.org:8080/cache/objects/9040/l2h/img15.png "$\displaystyle [Iinmathcal{F},Jinmathcal{I}(A)textrm{ and }forall xin I,(J:x)inmathcal{F}]Rightarrow Jinmathcal{F} $"){kind=small}