Radical of a ring

In ring theory, a branch of mathematics, a radical of a ring is an ideal of "bad" elements of the ring.

The first example of a radical was the nilradical introduced in (Köthe 1930), based on a suggestion in (Wedderburn 1908). In the next few years several other radicals were discovered, of which the most important example is the Jacobson radical. The general theory of radicals was defined independently by (Amitsur 1952, 1954, 1954b) and Kurosh (1953).

In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have an identity element. In particular, every ideal in a ring is also a ring.

A radical class (also called radical property or just radical) is a class σ of rings possibly without identities, such that:

The study of such radicals is called torsion theory.

For any class δ of rings, there is a smallest radical class Lδ containing it, called the lower radical of δ. The operator L is called the lower radical operator.

A class of rings is called regular if every non-zero ideal of a ring in the class has a non-zero image in the class. For every regular class δ of rings, there is a largest radical class Uδ, called the upper radical of δ, having zero intersection with δ. The operator U is called the upper radical operator.

A class of rings is called hereditary if every ideal of a ring in the class also belongs to the class.

Let R be any ring, not necessarily commutative. The Jacobson radical of R is the intersection of the annihilators of all simple right R-modules.

There are several equivalent characterizations of the Jacobson radical, such as:

As with the nilradical, we can extend this definition to arbitrary two-sided ideals I by defining J(I) to be the preimage of J(R/I) under the projection map R→R/I.

If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.

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