Editing Radical of a ring (section)

===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]].