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Zero divisor
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== Zero divisor on a module == Let {{mvar|R}} be a commutative ring, let {{mvar|M}} be an {{mvar|R}}-[[Module (mathematics)|module]], and let {{mvar|a}} be an element of {{mvar|R}}. One says that {{mvar|a}} is '''{{mvar|M}}-regular''' if the "multiplication by {{mvar|a}}" map <math>M \,\stackrel{a}\to\, M</math> is injective, and that {{mvar|a}} is a '''zero divisor on {{mvar|M}}''' otherwise.<ref name=Matsumura-p12>{{citation |author=Hideyuki Matsumura |author-link=Hideyuki Matsumura |year=1980 |title=Commutative algebra, 2nd edition |publisher=The Benjamin/Cummings Publishing Company, Inc. |page=12}}</ref> The set of {{mvar|M}}-regular elements is a [[multiplicative set]] in {{mvar|R}}.<ref name=Matsumura-p12/> Specializing the definitions of "{{mvar|M}}-regular" and "zero divisor on {{mvar|M}}" to the case {{math|1=''M'' = ''R''}} recovers the definitions of "regular" and "zero divisor" given earlier in this article.
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