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Difference between revisions of "Primitive ideal"

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''right primitive ideal''
 
''right primitive ideal''
  
A two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745801.png" /> of an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745802.png" /> (cf. [[Associative rings and algebras|Associative rings and algebras]]) such that the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745803.png" /> is a (right) [[Primitive ring|primitive ring]]. Analogously, by using left primitive rings one can define left primitive ideals. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745804.png" /> of all primitive ideals of a ring, endowed with some topology, is useful in the study of various classes of rings. Usually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745805.png" /> is topologized using the following [[Closure relation|closure relation]]:
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A two-sided ideal $P$ of an associative ring $R$ (cf. [[Associative rings and algebras]]) such that the quotient ring $R/P$ is a (right) [[primitive ring]]. Analogously, by using left primitive rings one can define left primitive ideals. The set $\mathfrak P$ of all primitive ideals of a ring, endowed with some topology, is useful in the study of various classes of rings. Usually $\mathfrak P$ is topologized using the following [[closure relation]]:
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$$
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\mathop{Cl} A = \left\{ { Q \in \mathfrak P : Q \supseteq \cap\{P:P\in A\} } \right\}
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$$
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where $A$ is a subset of $\mathfrak P$. The set of all primitive ideals of a ring endowed with this topology is called the ''[[structure space]]'' of this ring.
  
<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/p/p074/p074580/p0745806.png" /></td> </tr></table>
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR>
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</table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745807.png" /> is a subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745808.png" />. The set of all primitive ideals of a ring endowed with this topology is called the structure space of this ring.
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR></table>
 

Revision as of 20:47, 1 October 2016

right primitive ideal

A two-sided ideal $P$ of an associative ring $R$ (cf. Associative rings and algebras) such that the quotient ring $R/P$ is a (right) primitive ring. Analogously, by using left primitive rings one can define left primitive ideals. The set $\mathfrak P$ of all primitive ideals of a ring, endowed with some topology, is useful in the study of various classes of rings. Usually $\mathfrak P$ is topologized using the following closure relation: $$ \mathop{Cl} A = \left\{ { Q \in \mathfrak P : Q \supseteq \cap\{P:P\in A\} } \right\} $$ where $A$ is a subset of $\mathfrak P$. The set of all primitive ideals of a ring endowed with this topology is called the structure space of this ring.

References

[1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
How to Cite This Entry:
Primitive ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_ideal&oldid=39352
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article