Difference between revisions of "Modular ideal"
From Encyclopedia of Mathematics
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| − | 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 [[ | + | 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]] 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> | ||
Latest revision as of 16:16, 11 September 2016
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=39099
Modular ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_ideal&oldid=39099
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article