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| − | ''left, of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125601.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125602.png" />'' | + | ''left, of a set $X$ in $R$'' |
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| − | The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125603.png" /> of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125604.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125605.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125606.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125607.png" /> is a ring or a semi-group (or, generally, a groupoid) with a zero. The right annihilator of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125608.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a0125609.png" /> is defined in a similar manner as the set | + | The set $\mathfrak{Z}_l(X)$ of all elements $y$ in $R$ such that $yX = \{0\}$. Here $R$ is a ring or a semi-group (or, generally, a groupoid) with a zero. The right annihilator of a set $X$ in $R$ is defined in a similar manner as the set |
| − | | + | $$ |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256010.png" /></td> </tr></table>
| + | \mathfrak{Z}_r(X) = \{ z \in R : zy = \{0\} \} \ . |
| | + | $$ |
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| | The set | | The set |
| | + | $$ |
| | + | \mathfrak{Z}(X) = \mathfrak{Z}_l(X) \cap \mathfrak{Z}_r(X) |
| | + | $$ |
| | + | is the two-sided annihilator of $X$. In an associative ring (or semi-group) $R$ the left annihilator of an arbitrary set $X$ is a left ideal, and if $X$ is a left ideal of $R$, then $\mathfrak{Z}_l(X)$ is a two-sided ideal of $R$; in the non-associative case these statements are usually not true. |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256011.png" /></td> </tr></table>
| + | {{TEX|done}} |
| − | | |
| − | is the two-sided annihilator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256012.png" />. In an associative ring (or semi-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256013.png" /> the left annihilator of an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256014.png" /> is a left ideal, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256015.png" /> is a left ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256017.png" /> is a two-sided ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012560/a01256018.png" />; in the non-associative case these statements are usually not true.
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Revision as of 12:19, 1 November 2014
left, of a set $X$ in $R$
The set $\mathfrak{Z}_l(X)$ of all elements $y$ in $R$ such that $yX = \{0\}$. Here $R$ is a ring or a semi-group (or, generally, a groupoid) with a zero. The right annihilator of a set $X$ in $R$ is defined in a similar manner as the set
$$
\mathfrak{Z}_r(X) = \{ z \in R : zy = \{0\} \} \ .
$$
The set
$$
\mathfrak{Z}(X) = \mathfrak{Z}_l(X) \cap \mathfrak{Z}_r(X)
$$
is the two-sided annihilator of $X$. In an associative ring (or semi-group) $R$ the left annihilator of an arbitrary set $X$ is a left ideal, and if $X$ is a left ideal of $R$, then $\mathfrak{Z}_l(X)$ is a two-sided ideal of $R$; in the non-associative case these statements are usually not true.
How to Cite This Entry:
Annihilator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Annihilator&oldid=34139
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article