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.

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

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, $\mathcal{J}$.