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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" />''
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{{TEX|done}}
  
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
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''left, of a set $X$ in $R$''
  
<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>
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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
 +
$$
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\mathfrak{Z}_r(X) = \{ z \in R : Xz = \{0\} \} \ .
 +
$$
  
 
The set
 
The set
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$$
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\mathfrak{Z}(X) = \mathfrak{Z}_l(X) \cap \mathfrak{Z}_r(X)
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$$
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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.
  
<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>
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===Modules===
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Let $M$ be a left module over a ring $R$, and $X$ a subset of $M$.  The left annihilator of $X$ is 
 +
$$
 +
\mathfrak{Z}_l(X) = \{ z \in R : Xz = \{0\} \} \ .
 +
$$
 +
Again, the left annihilator of an arbitrary set $X$ is a left ideal. The annihilator of an element $x \in M$ is the annihilator of $\{ x \}$. As left $R$-modules we have
 +
$$
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R/\mathfrak{Z}_l(\{x\}) \cong Rx \ .
 +
$$
  
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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====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Algebra I" , Springer  (1998) {{ISBN|3-540-64243-9}}</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Algebra II" , Springer  (2003) {{ISBN|3-540-00706-7}}</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Lang, "Algebra" , Springer  (2002) {{ISBN|0-387-95385-X}}</TD></TR>
 +
</table>
 +
 
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===Linear spaces===
 +
Let $V$ be a [[vector space]] over a field $K$ and $V^*$ the dual space of [[linear functional‎]]s on $V$. For a subset $X$ of $V$, the annihilator
 +
$$
 +
X^\circ = \{ f \in V^* : f(X) = \{0\} \} \ .
 +
$$
 +
The annihilator of a general set $X$ is a subspace of $V^*$ and if $\langle X \rangle$ is the subspace of $V$ generated by $X$, then $X^\circ = \langle X \rangle^\circ$.

Latest revision as of 17:14, 15 November 2023


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 $$ \mathfrak{Z}_r(X) = \{ z \in R : Xz = \{0\} \} \ . $$

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.

Modules

Let $M$ be a left module over a ring $R$, and $X$ a subset of $M$. The left annihilator of $X$ is $$ \mathfrak{Z}_l(X) = \{ z \in R : Xz = \{0\} \} \ . $$ Again, the left annihilator of an arbitrary set $X$ is a left ideal. The annihilator of an element $x \in M$ is the annihilator of $\{ x \}$. As left $R$-modules we have $$ R/\mathfrak{Z}_l(\{x\}) \cong Rx \ . $$

References

[a1] N. Bourbaki, "Algebra I" , Springer (1998) ISBN 3-540-64243-9
[a2] N. Bourbaki, "Algebra II" , Springer (2003) ISBN 3-540-00706-7
[a3] S. Lang, "Algebra" , Springer (2002) ISBN 0-387-95385-X

Linear spaces

Let $V$ be a vector space over a field $K$ and $V^*$ the dual space of linear functional‎s on $V$. For a subset $X$ of $V$, the annihilator $$ X^\circ = \{ f \in V^* : f(X) = \{0\} \} \ . $$ The annihilator of a general set $X$ is a subspace of $V^*$ and if $\langle X \rangle$ is the subspace of $V$ generated by $X$, then $X^\circ = \langle X \rangle^\circ$.

How to Cite This Entry:
Annihilator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Annihilator&oldid=17045
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article