# 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. 1.
2. 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. 3.

$\mathcal{X}(R/\mathcal{X}(R))=0$.

The ideal $\mathcal{X}(R)$ is called the $\mathcal{X}$-radical  of $R$. A ring is called $\mathcal{X}$-radical if $\mathcal{X}(R)=R$, and is called $\mathcal{X}$-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 $\mathcal{X}$-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 well-known 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.

Quasi-regularity is a radical property. The associated radical is the Jacobson radical  , $\mathcal{J}$.

 Title radical theory Canonical name RadicalTheory Date of creation 2013-03-22 13:13:02 Last modified on 2013-03-22 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