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Primitive ideal

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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://www.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