Let 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 , the statement “ has property ” is either true or false.

We say that a ring which has the property is an -ring. An ideal of a ring is called an -ideal if, as a ring, it is an -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 is a radical property if it satisfies:

1. The class of -rings is closed under homomorphic images.
2. Every ring has a largest -ideal, which contains all other -ideals of . This ideal is written .
3. .

The ideal is called the -radical of . A ring is called -radical if , and is called -semisimple if .

If is a radical property, then the class of -rings is also called the class of -radical rings.

The class of -radical rings is closed under ideal extensions. That is, if is an ideal of , and and are -radical, then so is .

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 and such that isomorphic simple rings are in the same class, gives rise to a radical with the property that all rings in are -radical and all rings in are -semisimple. In fact, there are at least two distinct radicals for each such partition.

A radical is hereditary if every ideal of an -radical ring is also -radical.

A radical is supernilpotent if the class of -rings contains all nilpotent rings.

Examples

Nil is a radical property. This property defines the nil radical, .

Nilpotency is not a radical property.

Quasi-regularity is a radical property. The associated radical is the Jacobson radical, .