Difference between revisions of "Modular ideal"
From Encyclopedia of Mathematics
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| − | A right (left) [[Ideal|ideal]] | + | {{TEX|done}} |
| + | A right (left) [[Ideal|ideal]] $J$ of a ring $R$ having the following property: There is at least one element $e$ in $R$ such that for all $x$ from $R$ the difference $x-ex$ belongs to $J$ (respectively, $x-xe\in J$). The element $e$ is called a left (right) identity modulo the ideal $J$. In a ring with identity every ideal is modular. Every proper modular right (left) ideal can be imbedded in a maximal right (left) ideal, which is automatically modular. The intersection of all maximal modular right ideals of an associative ring coincides with the intersection of all maximal left modular ideals and is the [[Jacobson radical|Jacobson radical]] of the ring. Modular ideals are also called regular ideals. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR></table> | ||
Revision as of 18:11, 11 April 2014
A right (left) ideal $J$ of a ring $R$ having the following property: There is at least one element $e$ in $R$ such that for all $x$ from $R$ the difference $x-ex$ belongs to $J$ (respectively, $x-xe\in J$). The element $e$ is called a left (right) identity modulo the ideal $J$. In a ring with identity every ideal is modular. Every proper modular right (left) ideal can be imbedded in a maximal right (left) ideal, which is automatically modular. The intersection of all maximal modular right ideals of an associative ring coincides with the intersection of all maximal left modular ideals and is the Jacobson radical of the ring. Modular ideals are also called regular ideals.
References
| [1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |
How to Cite This Entry:
Modular ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_ideal&oldid=17576
Modular ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_ideal&oldid=17576
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article