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Difference between revisions of "Primitive ideal"

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''right primitive ideal''
 
''right primitive ideal''
  
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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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A two-sided ideal $ P $ of an [[Associative rings and algebras|associative ring]] $ R $ 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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\forall A \subseteq \mathfrak{P}: \qquad
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\operatorname{Cl}(A) \stackrel{\text{df}}{=} \left\{ Q \in \mathfrak{P} ~ \middle| ~ Q \supseteq \bigcap_{P \in A} P \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.
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The set of all primitive ideals of a ring endowed with this topology is called the '''[[structure space]]''' of this ring.
  
 
====References====
 
====References====
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<table>
 
<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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<TR><TD valign="top">[1]</TD><TD valign="top">
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N. Jacobson, “Structure of rings”, Amer. Math. Soc. (1956).</TD></TR>
 
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</table>
  
 
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Latest revision as of 04:53, 24 April 2017

right primitive ideal

A two-sided ideal $ P $ of an associative ring $ R $ 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: $$ \forall A \subseteq \mathfrak{P}: \qquad \operatorname{Cl}(A) \stackrel{\text{df}}{=} \left\{ Q \in \mathfrak{P} ~ \middle| ~ Q \supseteq \bigcap_{P \in A} P \right\}. $$ 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=41202
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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