View source for Primitive ideal

''right primitive ideal''

A two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745801.png" /> of an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745802.png" /> (cf. [[Associative rings and algebras|Associative rings and algebras]]) such that the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745803.png" /> is a (right) [[Primitive ring|primitive ring]]. Analogously, by using left primitive rings one can define left primitive ideals. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745804.png" /> of all primitive ideals of a ring, endowed with some topology, is useful in the study of various classes of rings. Usually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745805.png" /> is topologized using the following [[Closure relation|closure relation]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745806.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745807.png" /> is a subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074580/p0745808.png" />. 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>