# Primitive ideal

*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