Definition:Jacobson Radical

Definition

Let $A$ be a commutative ring with unity.

Let $\operatorname{maxspec} \left({A}\right)$ be the set of maximal ideals of $A$.

Then the Jacobson radical of $A$ is:

$\displaystyle \operatorname{Jac} \left({A}\right) = \bigcap_{\mathfrak m \in \operatorname{maxspec} \left({A}\right)}\mathfrak m$

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

Alternative notation

Some sources use $J \left({A}\right)$.

Source of Name

This entry was named for Nathan Jacobson.