The first example of a radical was the nilradical introduced by Köthe (1930), based on a suggestion of Wedderburn (1908).
The general theory of radicals was defined independently by (Amitsur 1952, 1954, 1954b) and Kurosh (1953).
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 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.
Every element of the Baer radical is nilpotent, so it is a nil ideal.
For commutative rings, this is just the nilradical and closely follows the definition of the radical of an ideal.
The set of left singular elements of a ring R is a two-sided ideal, called the left singular ideal, and is denoted
The von Neumann regular rings form a radical class.
The definition is left-right symmetric, and indeed produces a two-sided ideal of the ring.
This radical is important in the study of Noetherian rings, as outlined by Chatters & Hajarnavis (1980).