You are here
Homeupper nilradical
Primary tabs
upper nilradical
The upper nilradical $\operatorname{Nil}^{*}(R)$ of $R$ is the sum of all (twosided) nil ideals in $R$. In other words, $a\in Nil^{*}R$ iff $a$ can be expressed as a (finite) sum of nilpotent elements.
It is not hard to see that $\operatorname{Nil}^{*}(R)$ is the largest nil ideal in $R$. Furthermore, we have that $\operatorname{Nil}_{*}(R)\subseteq\operatorname{Nil}^{*}(R)\subseteq J(R)$, where $\operatorname{Nil}_{*}(R)$ is the lower radical or prime radical of $R$, and $J(R)$ is the Jacobson radical of $R$.
Remarks.

If $R$ is commutative, then $\operatorname{Nil}_{*}(R)=\operatorname{Nil}^{*}(R)=\operatorname{Nil}(R)$, the nilradical of $R$, consisting of all nilpotent elements.
Type of Math Object:
Definition
Major Section:
Reference
Parent:
Groups audience:
Mathematics Subject Classification
16N40 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections