Definition:Jacobson Radical

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Definition

Let $R$ be a commutative ring with unity.

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


Then the Jacobson radical of $R$ is:

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

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


Also denoted as

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


Source of Name

This entry was named for Nathan Jacobson.


Sources