Let $\xrad$ 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 $\xrad$ ' is either true or false.

We say that a ring which has the property $\xrad$ is an $\xrad$ ring. An ideal $I$ of a ring $R$ is called an $\xrad$ ideal if, as a ring, it is an $\xrad$ 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 $\xrad$ is a radical property if it satisfies:

1. The class of $\xrad$ rings is closed under homomorphic images.
2. Every ring $R$ has a largest $\xrad$ ideal, which contains all other $\xrad$ ideals of $R$ This ideal is written $\xrad(R)$
3. $\xrad(R/\xrad(R)) = 0$

The ideal $\xrad(R)$ is called the $\xrad$ radical of $R$ A ring is called $\xrad$ radical if $\xrad(R) = R$ and is called $\xrad$ semisimple if $\xrad(R) = 0$

If $\xrad$ is a radical property, then the class of $\xrad$ rings is also called the class of $\xrad$ radical rings.

The class of $\xrad$ radical rings is closed under ideal extensions. That is, if $A$ is an ideal of $R$ and $A$ and $R/A$ are $\xrad$ 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 $\xrad$ with the property that all rings in $\mathcal{R}$ are $\xrad$ radical and all rings in $\mathcal{S}$ are $\xrad$ semisimple. In fact, there are at least two distinct radicals for each such partition.

A radical $\xrad$ is hereditary if every ideal of an $\xrad$ radical ring is also $\xrad$ radical.

A radical $\xrad$ is supernilpotent if the class of $\xrad$ rings contains all nilpotent rings.

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, $\mathcal{J}$