# 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 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. | ||

− | <table | + | ====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}} | |

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## 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=16446