Jacobson Radical

For a ring R (usually with 1), the Jacobson radical J(R) is the intersection of all maximal left ideals of R. Equivalently, it is the set of elements that annihilate all simple left R-modules, or the largest quasi-regular ideal under suitable definitions.

For an associative unital ring R, the Jacobson radical J(R) is the intersection of all maximal left ideals (equivalently all maximal right ideals), equivalently the set of elements that annihilate every simple left R-module; in many algebraic contexts it is the largest quasi-regular ideal and for Artinian rings it is nilpotent.