Definition:Jacobson Radical
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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
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