Definition:Jacobson Radical

Definition
Let $R$ be a commutative ring with unity.

Let $\map {\operatorname{maxspec}} R$ be the set of maximal ideals of $R$.

Then the Jacobson radical of $R$ is:
 * $\ds \map {\operatorname {Jac} } R = \bigcap_{m \mathop \in \map {\operatorname{maxspec}} R} m$

That is, it is the intersection of all maximal ideals of $R$.

Also denoted as
Some sources use $\map J R$.

Also see

 * Definition:Nilradical of Ring