Jacobson radical

Remarks

Equivalent form: $x \in J(R)$ if and only if $1 - xy$ is a unit of $R$ for every $y \in R$. Example: in $\Z/12\Z$ the maximal ideals are $2\Z/12\Z$ and $3\Z/12\Z$, so $J(\Z/12\Z) = 6\Z/12\Z = \set{0, 6}$; for $y \in \Z/12\Z$, $1 - 6y$ is $1$ or $7$, both units. Non-example: $2 \notin J(\Z)$, since $2 \notin 3\Z$; with the equivalent form, $1 - 2 \cdot 2 = -3$ is not a unit of $\Z$. The nilradical is contained in $J(R)$, since every maximal ideal is prime. The inclusion can be strict: for $\Z_{(p)}$, the integers localized at a prime $p$, the nilradical is $0$ while $J = p\Z_{(p)}$. $R$ is a local ring exactly when $J(R)$ is a maximal ideal. Other notations are $\operatorname{Jac}(R)$ and $\operatorname{rad}(R)$. For an ideal $I$, the Jacobson radical of $I$ is the intersection of the maximal ideals that contain $I$.