Jacobson Radical as Intersection of Maximal Ideals

Statement

Let $R$ be a commutative ring. The Jacobson radical $J(R)$ is given by

the intersection of all maximal ideals of $R$.

A useful equivalent characterization is:

where “unit” means invertible element .

In particular, if $R$ is a local ring with maximal ideal $\mathfrak m$, then $J(R)=\mathfrak m$.

(Existence of maximal ideals in nonzero commutative rings uses existence of maximal ideals .)

Examples

1. Fields.
   If $R=k$ is a field, the only maximal ideal is $(0)$. Hence

   $$J(k)=(0).$$

2. A local ring.
   Let $R=k[x]/(x^2)$. This is local with maximal ideal $\mathfrak m=(\bar x)$. Therefore

   $$J(R)=\mathfrak m=(\bar x).$$

3. The integers.
   In $R=\mathbb Z$, the maximal ideals are $(p)$ for primes $p$. Their intersection is $(0)$, so

   $$J(\mathbb Z)=(0).$$

   (Similarly, in $\mathbb Z/n\mathbb Z$, $J(\mathbb Z/n\mathbb Z)$ is the ideal generated by the product of the distinct primes dividing $n$.)