# Difference between revisions of "Primitive ideal"

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''right primitive ideal'' | ''right primitive ideal'' | ||

− | A two-sided ideal | + | 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]]: |

+ | $$ | ||

+ | \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==== | |

− | + | <table> | |

+ | <TR><TD valign="top">[1]</TD><TD valign="top"> | ||

+ | N. Jacobson, “Structure of rings”, Amer. Math. Soc. (1956).</TD></TR> | ||

+ | </table> | ||

− | + | {{TEX|done}} | |

− |

## 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=16446