# Difference between revisions of "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.

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