Difference between revisions of "Primitive ideal"
From Encyclopedia of Mathematics
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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
Primitive ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_ideal&oldid=16446
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article