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

Source of Name

This entry was named for Nathan Jacobson.

Sources

• 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra: Chapter $1$: Rings and Ideals: $\S$ Nilradical and Jacobson Radical