Editing Radical of a ring (section)

==Examples==

===The Jacobson radical===
: {{main|Jacobson radical}}

Let ''R'' be any ring, not necessarily commutative.  The '''Jacobson radical of''' '''''R''''' is the intersection of the [[annihilator (ring theory)|annihilators]] of all [[simple module|simple]] right ''R''-modules.

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

*J(''R'') is the intersection of the regular [[maximal ideal|maximal]] right (or left) ideals of ''R''.
*J(''R'') is the intersection of all the right (or left) [[primitive ideal]]s of ''R''.
*J(''R'') is the maximal right (or left) quasi-regular right (resp. left) ideal of ''R''.

As with the [[nilradical of a ring|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'' &rarr; ''R/I''.

If ''R'' is commutative, the Jacobson radical always contains the nilradical. If the ring ''R'' is a [[finitely generated algebra|finitely generated]] '''Z'''-[[algebra over a ring|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]].

===The Baer radical===
The Baer radical of a ring is the intersection of the [[prime ideal]]s 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<sub>∗</sub>''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 of a ring|nilradical]] and closely follows the definition of the [[radical of an ideal]].

===The upper nil radical or Köthe radical===
The sum of the [[nil ideal]]s of a ring ''R'' is the upper nilradical Nil<sup>*</sup>''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.

=== Singular radical ===
An element of a (possibly [[non-commutative ring|non-commutative]] ring) is called left '''singular''' if it annihilates an [[essential submodule|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 [[singular submodule|left singular ideal]], and is denoted <math>\mathcal{Z}(_R R)</math>.  The ideal ''N'' of ''R'' such that <math>N/\mathcal{Z}(_R R)=\mathcal{Z}(_{R/\mathcal{Z}(_R R)} R/\mathcal{Z}(_R R))\,</math> is denoted by <math>\mathcal{Z}_2(_R R)</math> 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===
The Levitzki radical is defined as the largest [[locally nilpotent ideal]], analogous to the [[Hirsch–Plotkin radical]] in the theory of [[group (mathematics)|groups]].  If the ring is [[Noetherian ring|Noetherian]], then the Levitzki radical is itself a nilpotent ideal, and so is the unique largest left, right, or two-sided nilpotent ideal.{{cn|reason=Who introduced this notion? Is it named after Jacob Levitzki (1904-1956)?|date=October 2022}}

===The Brown–McCoy radical===
The Brown–McCoy radical (called the '''strong radical''' in the theory of [[Banach algebra]]s) can be defined in any of the following ways: 
* the intersection of the maximal two-sided ideals
* the intersection of all maximal modular ideals
* the upper radical of the class of all [[simple ring]]s with multiplicative identity

The Brown–McCoy radical is studied in much greater generality than associative rings with 1.

===The von Neumann regular radical===
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===
The Artinian radical is usually defined for two-sided [[Noetherian ring]]s as the sum of all right ideals that are [[Artinian module]]s.  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 {{harvtxt|Chatters|Hajarnavis|1980}}.
<!-- ===The Thierrin radical===
The Thierrin radical is the upper radical of all [[division ring]]s. It is hereditary.

===The Jenkins radical===
The Jenkins radical is the upper radical of all simple prime rings.
-->