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Difference between revisions of "Zero divisor"

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''in a ring or a semi-group with zero element''
 
''in a ring or a semi-group with zero element''
  
A non-zero element such that the product with some non-zero element is zero. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992301.png" /> is called a left (right) divisor of zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992302.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992303.png" />) for at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992304.png" />.
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A non-zero element such that the product with some non-zero element is zero. An element $a$ is called a left (right) divisor of zero if $ab=0$ ($ba=0$) for at least one $b\neq0$.
  
  
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992305.png" /> be a ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992306.png" /> a left module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992307.png" />. Then an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992308.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z0992309.png" /> is called a zero divisor in the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z09923010.png" /> if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z09923011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099230/z09923012.png" />.
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Let $A$ be a ring and $M$ a left module over $A$. Then an element $a\neq0$ of $A$ is called a zero divisor in the module $M$ if there is an $m\in M$ such that $am=0$.

Latest revision as of 12:22, 9 April 2014

in a ring or a semi-group with zero element

A non-zero element such that the product with some non-zero element is zero. An element $a$ is called a left (right) divisor of zero if $ab=0$ ($ba=0$) for at least one $b\neq0$.


Comments

Let $A$ be a ring and $M$ a left module over $A$. Then an element $a\neq0$ of $A$ is called a zero divisor in the module $M$ if there is an $m\in M$ such that $am=0$.

How to Cite This Entry:
Zero divisor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zero_divisor&oldid=17967
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article