radical theory
Let $\mathcal{X}$ represent a property which a ring may or may not have. This property may be anything at all: what is important is that for any ring $R$, the statement “$R$ has property $\mathcal{X}$” is either true or false.
We say that a ring which has the property $\mathcal{X}$ is an $\mathcal{X}$ring. An ideal $I$ of a ring $R$ is called an $\mathcal{X}$ideal if, as a ring, it is an $\mathcal{X}$ring. (Note that this definition only makes sense if rings are not required to have identity elements^{}; otherwise and ideal is not, in general, a ring. Rings are not required to have an identity element in radical theory.)
The property $\mathcal{X}$ is a radical property if it satisfies:

1.
The class of $\mathcal{X}$rings is closed under^{} homomorphic images^{}.

2.
Every ring $R$ has a largest $\mathcal{X}$ideal, which contains all other $\mathcal{X}$ideals of $R$. This ideal is written $\mathcal{X}(R)$.

3.
$\mathcal{X}(R/\mathcal{X}(R))=0$.
The ideal $\mathcal{X}(R)$ is called the $\mathrm{X}$radical^{} of $R$. A ring is called $\mathrm{X}$radical if $\mathcal{X}(R)=R$, and is called $\mathrm{X}$semisimple^{} if $\mathcal{X}(R)=0$.
If $\mathcal{X}$ is a radical property, then the class of $\mathcal{X}$rings is also called the class of $\mathrm{X}$radical rings.
The class of $\mathcal{X}$radical rings is closed under ideal extensions. That is, if $A$ is an ideal of $R$, and $A$ and $R/A$ are $\mathcal{X}$radical, then so is $R$.
Radical theory is the study of radical properties and their interrelations. There are several wellknown radicals which are of independent interest in ring theory (See examples – to follow).
The class of all radicals is however very large. Indeed, it is possible to show that any partition of the class of simple rings^{} into two classes $\mathcal{R}$ and $\mathcal{S}$ such that isomorphic^{} simple rings are in the same class, gives rise to a radical $\mathcal{X}$ with the property that all rings in $\mathcal{R}$ are $\mathcal{X}$radical and all rings in $\mathcal{S}$ are $\mathcal{X}$semisimple. In fact, there are at least two distinct radicals for each such partition.
A radical $\mathcal{X}$ is hereditary if every ideal of an $\mathcal{X}$radical ring is also $\mathcal{X}$radical.
A radical $\mathcal{X}$ is supernilpotent if the class of $\mathcal{X}$rings contains all nilpotent rings.
1 Examples
Nil is a radical property. This property defines the nil radical, $\mathcal{N}$.
Nilpotency is not a radical property.
Quasiregularity is a radical property. The associated radical is the Jacobson radical^{}, $\mathcal{J}$.
Title  radical theory 
Canonical name  RadicalTheory 
Date of creation  20130322 13:13:02 
Last modified on  20130322 13:13:02 
Owner  mclase (549) 
Last modified by  mclase (549) 
Numerical id  10 
Author  mclase (549) 
Entry type  Definition 
Classification  msc 16N80 
Related topic  JacobsonRadical 
Defines  radical 
Defines  radical property 
Defines  semisimple 
Defines  hereditary 
Defines  hereditary radical 
Defines  supernilpotent 
Defines  supernilpotent radical 