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 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=39352
Primitive ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_ideal&oldid=39352
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article