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Ideal ring structure
In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not good" elements of the ring. The first example of a radical was the nilradical introduced by , based on a suggestion of 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 and .

The Baer radical of a ring is the intersection of the prime ideals of the ring R. Equivalently it is the smallest semiprime ideal in R. The Baer radical is the lower radical of the class of nilpotent rings. Also called the "lower nilradical" (and denoted Nil∗R), the "prime radical", and the "Baer McCoy radical". 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 sum of the nil ideals of a ring R is the upper nilradical Nil*R or Köthe radical and is the unique largest nil ideal of R. Köthe's conjecture asks whether any left nil ideal is in the nilradical.

An element of a (possibly non commutative ring) is called left singular if it annihilates an essential left ideal, that is, r is left singular if Ir = 0 for some essential left ideal I. The set of left singular elements of a ring R is a two sided ideal, called the left singular ideal, and is denoted The ideal N of R such that is denoted by and is called the singular radical or the Goldie torsion of R. The singular radical contains the prime radical (the nilradical in the case of commutative rings) but may properly contain it, even in the commutative case. However, the singular radical of a Noetherian ring is always nilpotent.

The Levitzki radical is defined as the largest locally nilpotent ideal, analogous to the Hirsch–Plotkin radical in the theory of groups. If the ring is Noetherian, then the Levitzki radical is itself a nilpotent ideal, and so is the unique largest left, right, or two sided nilpotent ideal.

A von Neumann regular ring is a ring A (possibly non commutative without multiplicative identity) such that for every a there is some b with a = aba. The von Neumann regular rings form a radical class. It contains every matrix ring over a division algebra, but contains no nil rings.

The Artinian radical is usually defined for two sided Noetherian rings as the sum of all right ideals that are Artinian modules. 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 .